Effective Kan fibrations in simplicial sets
EEFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
BENNO VAN DEN BERG AND ERIC FABER
Abstract.
We introduce the notion of an effective Kan fibration, a new math-ematical structure that can be used to study simplicial homotopy theory. Ourmain motivation is to make simplicial homotopy theory suitable for homotopytype theory. Effective Kan fibrations are maps of simplicial sets equipped witha structured collection of chosen lifts that satisfy certain non-trivial properties.This contrasts with the ordinary, unstructured notion of a Kan fibration. Weshow that fundamental properties of Kan fibrations can be extended to explicitconstructions on effective Kan fibrations. In particular, we give a constructive(explicit) proof showing that effective Kan fibrations are stable under push for-ward, or fibred exponentials. This is known to be impossible for ordinary Kanfibrations. We further show that effective Kan fibrations are local, or completelydetermined by their fibres above representables. We also give an (ineffective)proof saying that the maps which can be equipped with the structure of an ef-fective Kan fibration are precisely the ordinary Kan fibrations. Hence implicitly,both notions still describe the same homotopy theory. By showing that the ef-fective Kan fibrations combine all these properties, we solve an open problemin homotopy type theory. In this way our work provides a first step in giving aconstructive account of Voevodsky’s model of univalent type theory in simplicialsets.
Contents
1. Introduction 41.1. Main contribution and motivation 41.2. Related work 51.3. Fibrations as structure 61.4. Effective Kan fibrations 91.5. Summary of contents 111.6. Acknowledgements 12Chapter 1. Π-types from Moore paths 132. Preliminaries 14
Date : September 29, 2020. a r X i v : . [ m a t h . C T ] S e p EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 3
10. Hyperdeformation retracts in simplicial sets 10210.1. HDRS are effective cofibrations 10210.2. HDRs as internal presheaves 10210.3. A small double category of HDRs 10610.4. Naive Kan fibrations in simplicial sets 12111. Mould squares in simplicial sets 12711.1. Small mould squares 12711.2. Effective Kan fibrations in terms of “filling” 13412. Horn squares 13912.1. Effective Kan fibrations in terms of horn squares 13912.2. Local character and classical correctness 15713. Conclusion 160Appendices 162A. Axioms 163A.1. Moore structure 163A.2. Dominance 166B. Degenerate horn fillers are unique 167Bibliography 168
EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS Introduction
Main contribution and motivation.
This paper is the first step in a pro-gramme by the authors to redevelop the foundations of simplicial homotopy theory,in particular around the Kan-Quillen model structure on simplicial sets, in a moreeffective or “structured” style. Our motivation comes from homotopy type theory (HoTT) and Voevodsky’s construction of a model of HoTT in simplicial sets [KL18],which relies heavily on the existence and properties of the Kan-Quillen model struc-ture.Type theory refers to a family of formal systems which can act both as foundationsfor (constructive) mathematics and functional programming languages. Recently, ithas become apparent that there exist many connections between type theory on theone hand and homotopy theory and higher category theory on the other. Besides Vo-evodsky’s fundamental contributions, other key steps have been the groupoid modelby Hoffman and Streicher [HS98], the interpretation of Martin-L¨of’s identity typesin categories equipped with a weak factorisation system [AW09] and the proof thattypes in type theory carry the structure of an ∞ -groupoid [BG11]; [Lum10]. As aresult of these contributions, homotopy type theory has become an active area ofresearch which keeps on developing at a quick pace, with implications for both typetheory and homotopy theory. (For an overview, see the HoTT book [Pro13].)However, for type theory to fully benefit from the rich treasure chest of homotopytheory and higher category theory, a computational understanding of the relevantresults from these areas is crucial. Indeed, we would like to think of type theoryas a framework for computation. Then to fully exploit homotopy-theoretic ideas inthis framework, one must be able to computationally reduce them. So a naturalquestion is how constructive Voevodsky’s model in simplicial sets is, or the proofs ofthe properties of the Kan-Quillen model structure on which it relies.One fundamental obstacle with building Voevodsky’s model in simplicial sets in aconstructive framework was identified by Bezem, Coquand and Parmann [BCP15].To interpret Π-types in simplicial sets, one uses that the category of simplicial setsis locally cartesian closed: that is, that the pullback functor along any map has aright adjoint, which we call push forward. Since the type families are interpreted asKan fibrations in Voevodsky’s model, we need to show that Kan fibrations are closedunder push forward along Kan fibrations. This is true classically, but as the authorsof [BCP15] show, this result is unprovable constructively. We will refer to this as theBCP-obstruction and, given the importance of Π-types in type theory, it is quite aserious problem.That this problem is not insurmountable was shown by Gambino and Sattler[GS17]: the key idea here is to treat being a Kan fibration not as a property, but asstructure. Inspired by the work in HoTT on cubical sets, they define a structurednotion of a uniform Kan fibration and give a constructive proof that uniform Kanfibrations are closed under push forward. They also show that their definition is“classically correct” in that a map can be equipped with the structure of a uniform FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 5
Kan fibration if and only if it has the right lifting property against the horn inclusions(is a Kan fibration in the usual sense).In this paper we will introduce another solution to this problem: the effective Kanfibrations . The reason for introducing a new solution is that Gambino and Sattlerran into trouble with another type constructor: universes. Indeed, the only knownmethod for constructing universal fibrations in simplicial sets is via the Hofmann-Streicher construction [HS97], and this method can only be applied to notions offibred structure which are local (see Definition 2.3 below). The difficulty with theirapproach is that it seems hard to prove constructively that the notion of a uniformKan fibration is local, whereas we are able to do this for our notion of an effectiveKan fibration.So, to summarise, our main contribution is the introduction of the notion of aneffective Kan fibration, a structured notion of Kan fibration for which we will provethe following results:(1) Effective Kan fibrations are closed under push forward.(2) The notion of an effective Kan fibration is local.(3) Effective Kan fibrations have the right lifting property against horn inclusions.(4) A map which has the right lifting property against horn inclusions can beequipped with the structure of an effective Kan fibration.We will give constructive proofs of (1) – (3), whereas the proof of (4) will necessarilybe ineffective (due to the BCP-obstruction).Besides having a clear computational content, another advantage of constructiveproofs is that they can be internalised to arbitrary Grothendieck toposes (not just S ets ). In fact, our arguments here can be internalised in any elementary topos witha natural numbers object, or a suitable predicative analogue (say, a locally cartesianpretopos with a natural numbers object). For those who prefer to think in terms ofset theory, our arguments can be performed in (a subsystem of) Aczel’s constructiveset theory CZF , which in turn is a subsystem of classical ZF , Zermelo-Fraenkel settheory (without choice).But however this may be, we feel that laying too great an emphasis on the meta-mathematical aspects of our work may be misleading. The task of reworking someof the fundamental concepts in simplicial homotopy theory in a more explicit orstructured style is an interesting undertaking in itself, whatever one’s foundationalconvictions, and we hope that any homotopy theorists reading this work will come tosee it that way as well. Indeed, any mathematician who wishes to skip the occasionalfoundational aside on our part should feel free to do so, and can read this paper asjust another piece of new mathematics.1.2. Related work.
Besides the work of Gambino and Sattler we already mentioned,there are two strands of research with which our approach should be compared.In response to the BCP-obstruction, most researchers in HoTT have abandonedsimplicial sets and switched to cubical sets. In doing so people have managed to
EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS constructively prove the existence of a model structure and a model of HoTT incubical sets. In addition, their cubical models can be seen as interpreting a cubicaltype theory, in which principles like univalence can be derived, and which enjoys(homotopy) canonicity (see [Coh+17]; [CHS19]; [Hub19]; [BCH19]).These are impressive results and our approach is by no means that far advanced.However, we still feel that analogous results for simplicial sets would be preferable:indeed, simplicial techniques pervade modern homotopy theory, much more than cu-bical approaches do, and in order to connect to most of the ongoing work in homotopytheory and higher category theory, a simplicial approach is more likely to be success-ful. In addition, it is at present not entirely clear whether any of the constructivemodel structures that people have developed in cubical sets model the world of ho-motopy types or ∞ -groupoids.The other approach with which our work should be compared is that of Gambino,Henry, Sattler and Szumi(cid:32)lo, who, in face of the BCP-obstruction, decide to bite thebullet (see [Hen19]; [GH19]; [GSS19]). Their starting point was the constructive proofby Simon Henry of the existence of the Kan-Quillen model structure on simplicial sets,using the standard definitions of the Kan and trivial Kan fibrations (having the rightlifting property against the horn inclusions and boundary inclusions, respectively).Based on this work, Henry in collaboration with Gambino managed to constructa model of HoTT, modulo some tricky coherence problems. Their work has theadvantage that it is based on the usual definitions of the (trivial) Kan fibrations,so in that sense it looks, at least at first glance, more familiar than our structuredapproach. In addition, their work is definitely more advanced than ours.However, we still think that a structured approach looks more appealing. Due tothe BCP-obstruction, they only have a weak form of Π-types. In comparision, ourapproach should give us genuine Π-types with definitional η - and β -rules. Also, itseems that to obtain a genuine model of homotopy type theory based on their workforces one to solve some quite difficult coherence problems, for which at present nosolutions are known. In contrast, we expect that a more structured approach will behelpful in solving any coherence problems we would encounter if we would start toturn our work into a model of type theory.1.3. Fibrations as structure.
So what is our notion of an effective Kan fibration?Before we answer that question, let us first discuss what, in general, we mean by astructured notion of fibration.A common situation in homotopy theory is that we are working in some category E equipped with a pullback stable class of fibrations; by pullback stability we meanthat in a pullback square like Y (cid:48) YX (cid:48) X p (cid:48) p FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 7 the map p (cid:48) will be a fibration whenever p is. If one conceptualises matters like this,being a fibration is a property of a map; in this paper, however, we will think of beinga fibration as structure . In particular, the setting of a category E with a pullbackstable class of fibrations will be replaced by a structured notion of a fibration or anotion of fibred structure on E , which we will define as a presheafFib: ( E → cart ) op → Sets, where E → cart denotes the category of arrows in E and pullback squares between them.Given such a structured notion of fibration, an element σ ∈ Fib( p ) will be called a fibration structure on p : Y → X ; and if such an element σ exists, we may call p afibration. These fibrations will form a pullback stable class, as before. Indeed, we canthink of a pullback stable class as a degenerate notion of fibred structure where Fib( p )always contains at most one element, signalling whether the map p is a fibration ornot.But in many examples being a fibration is quite naturally thought of as additionalstructure on a map. For instance, a common way of defining a class of fibrations isby saying that they are cofibrantly generated by a class of maps A . That is, a map p : Y → X is a fibration precisely when for any m : B → A ∈ A and commutativesquare B YA X fm pg there exists a dotted filler as shown making both triangles commute; one also saysthat p has the right lifting property against A . In this situation we can define astructured notion of fibration Fib by declaring the elements of Fib( p ) to be liftingstructures on p : that is, functions which assign to each square like the one above with m ∈ A a filler σ m,f,g : A → Y making both triangles commute. Let us for the momentwrite this notion of fibred structure as RLP( A ).One thing which happens if one shifts to a structured style is that notions offibration which are equivalent as properties are no longer isomorphic as structures.Take the trivial Kan fibrations in simplicial sets as an example. They can be definedas the maps which are cofibrantly generated by the monomorphisms; or as thosewhich are cofibrantly generated by the monomorphisms S ⊆ ∆ n with representablecodomain; or as those which are cofibrantly generated by the boundary inclusions ∂ ∆ n ⊆ ∆ n . These may all be equivalent as properties, but as structures, they are alldifferent. Indeed, there are “forgetful” morphisms of fibred structure (presheaves)RLP(monos) → RLP(sieves) → RLP(boundary inclusions) , but they are not monomorphisms, let alone isomorphisms. So as structured notionsof fibration they need to be carefully distinguished.This leads us to another important point for this paper: one can try to repair thisby imposing compatibility conditions on the lifting structure (also known as uniformityconditions in the literature). Indeed, in the way we have defined RLP( A ) its elements EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS σ choose solutions for a class of lifting problems, but there are no conditions sayinghow these solutions should be related. For instance, suppose we have in simplicialsets a solid diagram of the form D C YC A X n m pkl l in which p is a trivial Kan fibration and the left hand square is a pullback involvingmonomorphisms n and m . Then any element in RLP(monos)( p ) must, among otherthings, choose dotted arrows l and l as shown; we could define a notion of fibredstructure RLP c (monos) which would require that in such circumstances we must have k.l = l . And if RLP c (sieves) would be the restriction of RLP(sieves) to those liftingstructures which in a situation like α ∗ S S Y ∆ m ∆ n X n m pαl l would choose lifts l and l satisfying α.l = l , then the forgetful morphismRLP c (monos) → RLP c (sieves)would be an isomorphism of notions of fibred structure. The reader may wonder if anytrivial Kan fibration can still be equipped with such a structure: that is, whether liftsagainst monos or sieves can always be chosen in such a way that these compatibilityconditions are met. That is indeed the case (see [GS17]).Both RLP c (monos) and RLP c (sieves) are examples of right lifting structures de-fined by lifts against categories rather than classes of maps, and a rich variety oflifting structures is quite characteristic of our structured approach. Indeed, we willalso consider double categorical and even triple categorical notions of lifting structure.To motivate this, let us consider the forgetful mapRLP(monos) → RLP(boundary inclusions) . This will not be a monomorphism even when we restrict to RLP c (monos), but thereis a further (double-categorical) compatibility condition we could imagine imposingwhich would have this effect. Suppose we have a solid diagram in simplicial sets C YBA X gm pn f in which p is a trivial Kan fibration and m and n are monomorphisms. Then a liftingstructure σ on p will give rise to a filler A → Y in two different ways: we can use thatmonomorphisms are closed under composition and take σ n.m,g,f . But we could also FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 9 first construct a map l : B → Y by taking l = σ m,g,f.n and then use that to construct σ n,l,g . A natural requirement would be that these two lifts should always coincide. Ifwe write RLP dc (monos) for the notion of fibred structure where the lifts satisfy thiscondition on top of the previous one, then we will prove in this paper thatRLP dc (monos) → RLP(boundary inclusions) . is a monomorphism of notions of fibred structure. We will also characterise the imageof this map and show that every trivial Kan fibration can be equipped with such adouble-categorical lifting structure. Indeed, with one further (constructive) twist,this will be our preferred structured notion of a trivial Kan fibration (an effectivetrivial Kan fibration ).1.4. Effective Kan fibrations.
As said, the core of our paper is the definition ofan effective Kan fibration, our preferred structured notion of a Kan fibration. Tomotivate this definition, let us recall the classical result (from [GZ67]) that saysthat the Kan fibrations are cofibrantly generated by maps of the form m ˆ ⊗ ∂ i , where m : A → B is a cofibration, ˆ ⊗ is the pushout-product and ∂ i : 1 → I is one of the twoendpoint inclusions into the interval I = ∆ . This can be reformulated as follows:let us say that a map p : Y → X has the right lifting property against a commutativesquare D BC A if for any solid diagram
D B YC A X p and dotted arrow C → Y making the diagram commute, there exists a dotted arrow A → Y making the whole picture commute. Then a map is a Kan fibration if and onlyif it has the right lifting property against the left hand square in a double pullbackdiagram of the form A A × I AB B × I B. (1 ,∂ i ) m m × I π m (1 ,∂ i ) π The usual definition of a Kan fibration in terms of horn inclusions can also be statedas a lifting problem against a square, namely the left hand square in another doublepullback diagram: ∂ ∆ n s ∗ ∂ ∆ n ∂ ∆ n ∆ n ∆ n +1 ∆ n , d s where s = s i is one of the degeneracies and d = d i /d i +1 is one of its sections. We willcall such left hand squares horn squares .These two situations have something in common, namely that they are both liftingconditions against a left hand square in a double pullback diagram of the form C r ∗ C CA B A m mi r in which m is a cofibration and ( i, r ) is a deformation retract of some kind. The firststep towards our definition of an effective Kan fibration is the identification of theright kind of deformation retracts. Our solution is the notion of a hyperdeformationretract (HDR), and to define these HDRs we use the simplicial Moore path functordefined by the first author in collaboration with Richard Garner [BG12].Once we have the concept of an HDR, we can define the mould squares as thosepullback squares A (cid:48) B (cid:48) A B m ( i (cid:48) ,r (cid:48) )( i,r ) in which m is a cofibration, ( i, r ) and ( i (cid:48) , r (cid:48) ) are HDRs and the square (read from topto bottom) is what we will call a cartesian morphism of HDRs. The idea, then, isto define the effective Kan fibrations as those maps which come equipped with liftsagainst mould squares.What is missing from this definition, however, are the correct compatibility con-ditions. It turns out that mould squares can be composed both horizontally andvertically (they naturally fit into a double category), and this leads to two naturalcompatibility conditions. In fact, there is a further “perpendicular” condition, be-cause mould squares can be pulled back along morphisms of HDRs, leading to a third(triple-categorical) compatibility condition.Once we have this in place, and we have checked that horn squares are mouldsquares, it follows immediately that effective Kan fibrations have the right liftingproperty against horn inclusions (it will also not be too hard to see that our effectiveKan fibrations are uniform Kan fibrations in the sense of Gambino-Sattler). In fact,quite a lot of pages will be spent on proving that the lifts against the mould squaresare completely determined by the lifts against the horn squares, or, in other words,that the forgetful mapRLP tc (mould squares) → RLP(horn squares)is a monomorphism of fibred structures. We will also characterise its image, whichwill be crucial for proving both that our notion of an effective Kan fibration is localand that it is classically correct. The paper will be consist of two parts and these tworesults will form the main achievements of the second part of this paper.
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 11
The first part will be devoted to proving that the effective Kan fibrations areclosed under push forward. We find it convenient to do this axiomatically, using anaxiomatic setup reminiscent of the work of Orton and Pitts [OP18]. The idea ofOrton and Pitts (but see also [GS17]; [FB19]) was to develop the basic theory ofthe cubical sets model in the setting of a suitable category equipped with a class ofcofibrations forming a dominance and an interval object I . In our setup we will keepthe dominance, but replace the interval object by a Moore path functor M satisfyingcertain equations (these can be found in an appendix to this paper). The examplewe have in mind is, of course, the simplicial Moore path functor from [BG12]. Asour dominance, we take the monomorphisms in simplicial sets which are “pointwisedecidable” (this is an additional constructive requirement that we impose on thecofibrations, which can be ignored by our classical readers). As we will show, thisaxiomatic setting is sufficiently powerful to define a suitable notion of mould squareand effective Kan fibration, and prove that the effective Kan fibrations are closedunder push forward.1.5. Summary of contents.
The contents of this paper are therefore as follows.We start Part 1 with a recap of the theory of algebraic weak factorisation systems(AWFSs), a structured analogue of the notion of a weak factorisation system. In thisstructured notion the left maps are replaced by coalgebras for a comonad on the arrowcategory, while the right maps are replaced by the algebras for a monad on the arrowcategory. Our main reference for this theory is an important paper by Bourke andGarner [BG16], which also explains the connection to double categories. There aretwo (related) points here which are perhaps worth stressing for those who are alreadyfamiliar with this theory: first of all, for us the distributive law is important and wewill always assume it. Secondly, we will exclusively work with the (co)algebras forthe (co)monad, never with the (co)algebras for the (co)pointed endofunctor.We will then go on to explain how both dominances and Moore structures give riseto AWFSs (for dominances this can already be found in [BG16]). Anticipating whathappens in simplicial sets, we will refer to these as the (effective cofibration, effectivetrivial fibration) and (HDR, naive fibration)-AWFS, respectively. Using these two in-gredients we will then define the notions of mould square and effective Kan fibration.Assuming that the Moore structure is symmetric, we will then show that these effec-tive Kan fibrations are closed under pushforward. An important intermediate stepfor this is the proof of the Frobenius property for the (HDR, naive fibration)-AWFS,which is related to an argument that can also be found in [BG12].We will start the second part by showing that the category of simplicial sets can beequipped with both a dominance and a symmetric Moore structure. This will showthat the theory of part 1 applies to simplicial sets. Then we will proceed to showthat effective Kan fibrations can be completely characterised by their lifts againsthorn squares, which will prove both that this notion of fibred structure is local andclassically correct.
To our surprise it turns out that the machinery we develop here can also be usedto give effective (structured) analogues of the notions of left and right fibration insimplicial sets. Indeed, also these can be defined by a right lifting property againsta class of mould squares, using the same dominance of cofibrations, but a differentMoore structure. When our results have implications for an effective theory of leftand right fibrations, we will comment on that as well.Finally, we will finish this paper with a conclusion outlining directions for futureresearch and two appendices. In the first appendix we give our version of the Orton-Pitts axioms, while the second appendix proves a result on horn fillers that we needfor the proofs that our different effective notions of fibration are classically correct.1.6.
Acknowledgements.
We thank Richard Garner for some very useful conver-sations on polynomial functors, which had a major influence on Section 9.HAPTER 1 Π -types from Moore paths
134 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS Preliminaries
In this section we introduce the main theoretical framework in which our theoryof effective fibrations is embedded. Abstractly put, we are studying and constructingnew notions of fibred structure and cofibred structure on a category E . Throughoutthe paper, some basic conditions on E are always assumed to hold. First, we willassume that E has finite limits. Second, we will require that functor(1) cod: E → → E from the arrow category of E to E which sends arrows to their codomain is a bifibration satisfying the Beck-Chevalley condition (see Box 1.2 on page 47). Besides pullbacks(which we already have), this also requires the existence of arbitrary pushouts, to-gether with a compatibility condition between them. This condition is as follows.Given a commutative cube: C (cid:48) D (cid:48) C DA (cid:48) B (cid:48) A B such that(i) The bottom square
ABCD is a pullback;(ii) The right square B (cid:48) D (cid:48) BD is a pushout;(iii) The back square A (cid:48) B (cid:48) AB is a pullback;then the left square A (cid:48) C (cid:48) AC is a pushout if and only if the front square C (cid:48) D (cid:48) CD isa pullback. One can check that this condition is a weakening of E being cocompleteand locally cartesian closed. To summarise, we assume on E the following conditionsthroughout the first part of our paper:(i) It has finite limits and pushouts;(ii) Pullbacks and pushouts are compatible in that cod: E → → E is a bifibrationsatisfying the Beck-Chevalley condition.Note that the main result of the first part, contained in Section 6, can only be statedwhen E is actually locally cartesian closed. We will also require the existence of aninitial object and (hence) finite colimits in that section – so the conditions (i) and(ii) will be satisfied automatically in that case.2.1. Fibred structure.Definition 2.1.
Let E be a category with finite limits and write E → cart for the categoryof arrows in E and pullback squares between them. A presheaf fib on E → cart fib : ( E → cart ) op → S ets FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 15 will also be called a notion of fibred structure . A morphism of notions of fibred struc-ture is simply of a morphism of presheaves and and two notions of fibred structurewill be called equivalent if they are naturally isomorphic as presheaves.Lastly, a notion of cofibred structure refers to the dual, which is the same as apresheaf cofib : E → cocart → S ets where E → cocart denotes the category of arrows in E with pushout squares between them. Notation 2.2.
In this paper, we often have to deal with essentially the same struc-ture, but now as a fibred structure, then as a category, or even a double category.We distinguish between these three settings by denoting them as fib , Fib , and F ib respectively. Definition 2.3.
Let fib be a notion of fibred structure on a category E . We will callthe notion of fibred structure fib local (or locally representable ) if the following holdsfor any small diagram D : I → E → cart with colimit f and colimiting cocone ( σ i : Di → f : i ∈ Ob( I )):If we can choose fibration structures x i ∈ fib ( Di ) for any i ∈ Ob( I )such that fib ( Dα )( x i ) = x j for any α : j → i in I , then there exists aunique fibration structure x ∈ fib ( f ) such that fib ( Dσ i )( x ) = x i .We will mainly be interested in notions of fibred structure on presheaf categories(in fact, on the category of simplicial sets), in which case the notion of locality canbe defined in a different way. Proposition 2.4.
Suppose E is the category of presheaves on C , and let fib be anotion of fibred structure on E . Then fib is local if and only if the following holds forany morphism f : Y → X in E : if we choose for any x ∈ X ( C ) a fibration structure s x on a pullback f x : Y x → yC as in Y x YyC X f x fx such that for any x ∈ X ( C ) and α : D → C in C , the fibration structure s x on f x pulls back to the one chosen on f x · α for the pullback square Y x · α Y x yD yC, f x · α f x α over f , then there exists a unique fibration structure s ∈ fib ( f ) which pulls back to s x for any pullback square of the first type.Proof. The equivalence uses standard properties of presheaf categories. Every ob-ject is a colimit of representables, and since pullback along f preserves colimits, thecondition becomes a special case of Definition 2.3. For the other direction, one can reduce Definition 2.3 to the special case by takingpullbacks along representables: Y i,x Y i Yy C X i X σ i ⇒ x D ( i ) f Since cod: E → → E is a left adjoint, it preserves colimits, so every x (cid:48) : y C → X factorsthrough some X i since we are in a category of presheaves. Hence the fibred structuredetermines precisely the input data for the special case. The pullback property forthe unique induced fibration follows from the uniqueness condition on each of the D ( i ) that follows from the special case. (cid:3) Remark 2.5.
A local notion of fibred structure is a structured analogue of a localclass of maps as in [Sat18, Remark 4.4] and [Cis14, Definition 3.7], for instance. Anearlier structured analogue appears in Shulman’s paper [Shu19, Proposition 3.18].The definition we gave here is a simplification of his, because we demand that thefibration structures are strictly functorial under pullback, rather than pseudofuncto-rial.2.2.
Double categories of left and right lifting structures.
We recall the defi-nition of a double category : Definition 2.6. A double category A consists of:(i) A collection of objects together with two separate category (morphism) struc-tures on it, called horizontal and vertical morphisms.(ii) A special category structure whose objects are the vertical morphisms, andwhose arrows are called squares . The special property is that every square froma vertical morphism u to another vertical morphism v has a ‘pointwise’ domain,and a ‘pointwise’ codomain given by horizontal morphisms: f : dom u → dom v , g : cod u → cod v Moreover, composition of squares respects composition of these horizontal mor-phisms and identity squares have identity horizontal morphisms for their point-wise domain and codomain.Further, there is a ‘pointwise’ or ‘vertical’ composition operation of squareswith matching pointwise domains and codomains, which extends composition ofvertical morphisms. We often think of a square s : u → v as filling in a diagramof horizontal and vertical arrows: A BC D s ⇒ fgu v These diagrams can be composed both horizontally (ordinary composition ofsquares) and vertically (pointwise composition).
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 17
Alternatively, one can verify that a double category is the same thing as an internalcategory in the ‘category’ of large categories: L × L L L L ◦ domidcod where ‘ ◦ ’ denotes vertical, or pointwise, composition, whilst horizontal compositionis the composition of the two categories L (vertical morphisms and squares) and L (objects and horizontal morphisms).For double categories A , B , a double functor F : A → B is a compatible tripleof functors (which we can denote by the same F ) between categories of horizontalmorphisms, vertical morphisms, and squares, which in addition respects pointwisecomposition. Here compatible means that the image of a square as drawn abovelooks like: F ( A ) F ( B ) F ( C ) F ( D ) F ( f ) F ( g ) F ( u ) F ( v ) Again, one can verify that a double functor between double categories is the samething as an internal functor between the corresponding internal categories in thecategory of large categories.
Example 2.7.
The typical example of a double category is the category of arrowsof any category E . The horizontal and vertical arrows are both given by the categorystructure, and squares are given by commutative squares. We will denote this doublecategory by Sq ( E ).In this paper, we often work with double categories over the category of squaresof some category E , i.e. double functors A → Sq ( E ). For the next construction, webegin with such a double functor, denoted I : L → Sq ( E ). For a morphism p : Y → X in E , a right lifting structure with respect to I consists of:(i) A family φ − , − ( − ) of arrows in E consisting of the following. For every verticalmorphism v in L , and every commutative square in E , as in the solid part ofthe following diagram: A YB X fgI ( v ) pφ f,g ( v ) the arrow φ f,g ( v ): B → Y is a diagonal filler as drawn which makes the diagramcommute. A commutative square like the above is called a lifting problem .(ii) The condition that for every such φ f,g ( v ), and every square v (cid:48) → v whose imageunder I is given by the left-hand commutative square in the solid part of the diagram below: A (cid:48) A YB (cid:48)
B X I ( v (cid:48) ) I ( f (cid:48) ) I ( g (cid:48) ) I ( v ) fg pφ f.I ( f (cid:48) ) ,g.I ( g (cid:48) ) ( v (cid:48) ) φ f,g ( v ) ,the drawn morphisms make the diagram commute, i.e. φ f,g ( v ) .I ( g (cid:48) ) = φ f.I ( f (cid:48) ) ,g.I ( g (cid:48) ) ( v (cid:48) )(iii) The condition that when v, w is a composable pair of vertical arrows, i.e. cod v =dom w , we have: φ φ f,I ( w ) .g ( v ) ,g ( w ) = φ f,g ( w.v )In diagrammatic notation, this means that the two ways to fill the below dia-gram, either in two steps or in one go, are the same: A YA (cid:48)
B X fgI ( v ) I ( w ) p When ( u, v ): p (cid:48) → p is a pullback square as drawn in the diagram below, and φ is aright lifting structure on p with respect to I , the universal property of the pullbackinduces a right lifting structure ( u, v ) ∗ φ for p (cid:48) with respect to I : A Y (cid:48)
YB X (cid:48) X I ( v ) fg p (cid:48) uv p ( u, v ) ∗ φ φ u.f,v.g ( v ) ,It is easy to verify that this is a right lifting structure for p (cid:48) . This conclusion issummarised as follows: Proposition 2.8.
With this pullback action, there is, for every I : L → Sq ( E ) asabove, a fibred structure on E which sends each arrow to the set of right lifting struc-tures with respect to I on it. We denote this fibred structure by the functor: I (cid:116)(cid:116) ( − ): ( E → cart ) op → S ets (cid:3) FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 19
When ( q : Z → Y, φ ), ( q : Y → X, ψ ) are two composable arrows in E together witha right lifting structure with respect to I on them, there is a candidate right liftingstructure on the composition q.p defined by step-wise lifts: A ZYB X fgI ( v ) pqφ p.f,g ( v ) ψ f,φp.f,g ( v ) ( v ) To verify that this is a right-lifting structure, we have to verify two conditions. For(ii), this is: ψ f,φ p.f,g ( v ) ( v ) .I ( g (cid:48) ) = ψ f.I ( f (cid:48) ) ,φ p.f,g ( v ) .I ( g (cid:48) ) ( v (cid:48) )= ψ f.I ( f (cid:48) ) ,φ p.f.I ( f (cid:48) ) ,g.I ( g (cid:48) ) ( v (cid:48) ) ( v (cid:48) )For condition (iii), this is better done diagrammatically: A ZA (cid:48)
YB X I ( v ) I ( w ) pqfg ( a (cid:48) )( c (cid:48) )( b (cid:48) ) A ZA (cid:48)
YB X I ( v ) I ( w ) pqfg ( a )( b )( c ) ( d ) Consider the two diagrams above. On the left-hand side, a lift is obtained intwo steps ( a (cid:48) ) and ( c (cid:48) ), but we immediately note that the second step could havebeen done in two steps by first finding ( b (cid:48) ) and then finding the same ( c (cid:48) ). On theright-hand side, a lift is found in four steps ( a )-( d ). Note that( b (cid:48) ) = ψ f, ( a (cid:48) ) .I ( w ) ( v ) = ψ f, ( a ) ( v ) = ( b )And similarly:( c ) = φ p. ( b ) ,g ( w ) = φ ( a ) ,g ( w ) = φ φ p.f,I ( w ) .g ( v ) ,g ( w ) = φ p.f,g ( w.v ) = ( a (cid:48) )Therefore it follows that ( d ) and ( c (cid:48) ) are the same. So the composed lifting structureindeed satisfies the third property above. As a consequence, the following definitionis just: Definition 2.9.
Suppose I : L → Sq ( E ) is a double category over Sq ( E ). We candefine a new double category I (cid:116)(cid:116) : L (cid:116)(cid:116) → Sq ( E ) as follows:(i) Objects are objects of E , and horizontal morphisms are morphisms in E (ii) A vertical morphism Y → X is a pair ( p : Y → X, φ ) where φ is a right liftingstructure for p with respect to I ; that is, φ ∈ I (cid:116)(cid:116) ( p ) in the notation of Proposition2.8. Composition of vertical morphisms is defined as above. (iii) A square ( p (cid:48) , φ (cid:48) ) → ( p, φ (cid:48) ) between vertical morphisms is a commutative square p (cid:48) → p in E as on the right hand side in the diagram below, such that wheneverthere is a lifting problem: A Y (cid:48)
YB X (cid:48) X I ( v ) fg p (cid:48) kl pφ l.f,k.g ( v ) φ (cid:48) f,g ( v ) , the induced diagram as drawn commutes, that is: φ l.f,k.g ( v ) = l.φ (cid:48) f,g ( v ) . Note that it needs to be checked that vertical composition of squares is compatiblewith the composition operation on vertical morphisms, but this follows easily fromthe definition of a square.Similarly, there is a notion of left lifting structure for an arrow f : A → B withrespect to a double category J : R → Sq ( E ). This consists of a family of fillers forevery commutative square: A YB X fgf J ( v ) φ f,g ( v ) such that three analogous conditions (i)-(iii) hold. For the sake of brevity, we willnot repeat those here. In fact, the definition is completely dual to a right liftingstructure, as follows. Define the opposite of a double category A to be the doublecategory A op with horizontal and vertical arrows as well as squares reversed, e.g. Sq ( E ) op ∼ = Sq (cid:0) E op (cid:1) , with the obvious extension to double functors. Then a leftlifting structure for an arrow f with respect to J : R → Sq ( E ) is the same thing asa right lifting structure for f op : B → A in E op with respect to J op : R op → Sq (cid:0) E op (cid:1) .From Proposition 2.8, it follows that there is a functor (cid:116)(cid:116) J : E → cocart → S ets where E → cocart is the category of arrows in E and pushout squares between them, whichsends an arrow to the set of left lifting structures on it. In other words, left liftingstructure is a notion of cofibred structure (Definition 2.1). The following definitionfollows Definition 2.9: Definition 2.10.
Suppose J : R → Sq ( E ) is a double category over Sq ( E ). We candefine a new double category (cid:116)(cid:116) J : (cid:116)(cid:116) R → Sq ( E ) analogous to Definition 2.9, but wherethe vertical morphisms have a left lifting structure with respect to J .Recall that the category of arrows of a category E and squares between them isdenoted E → . This is the same as the category structure that exists on the verticalmorphisms in Sq ( E ). The following proposition allows us to define left and right FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 21 lifting structures for mere ‘categories of arrows’
A → E → as a special case in Definition2.12 below. In the following proposition, the category of (small) double categoriesand double functors is denoted by DBL . Proposition 2.11.
The functor ( − ) : DBL / Sq ( E ) → Cat / E → which takes a double functor I : L → Sq ( E ) to the category of vertical arrows andsquares I : L → E → over E → has a fully faithful left adjoint, i.e. it is a coreflection.Proof. Suppose I : L → E → is a functor. Then we can define a double category I dbl : L dbl → Sq ( E ) as follows:(i) Objects of L dbl are pairs ( i, v ) where v is an object of L and i ∈ { , } (ii) Horizontal arrows ( i, s ): ( i, u ) → ( j, v ) require that i = j and are given bymorphisms s : u → v in L (iii) The only non-identity vertical arrows are given by v : (0 , v ) → (1 , v )(iv) For every arrow s : u → v in L , there is a square s : 1 ( i,u ) → ( i,v ) for every i ∈ { , } , and a square s : u → v between the non-identity morphisms of (ii).Vertical composition of squares is trivial, in that s.s = s .Alternatively, it can be presented by the following internal category in the categoryof (large) categories:( L + L ) + ( L + L ) L + ( L + L ) L + L [[1 L , L ] , L + L ] [ inl , L + L ] inr [ inr , L + L ] The functor I dbl then sends a square s : u → v to: A BC D
I(s) I ( u ) I ( v ) It is easy to see that this construction is functorial and fully faithful. The unit istrivial, and the counit (cid:15) I : ( I ) dbl → I sends vertical arrows to vertical arrows, squaresto squares, and sends objects (0 , v ) to dom v and (1 , v ) to cod v , and the same forhorizontal morphisms and their pointwise domain/codomain. It is easy to see thatthis constitutes a counit and that ( − ) is a coreflection. (cid:3) Definition 2.12.
Suppose I : L → E → is a functor. Then we can define a new functor I (cid:116) : L (cid:116) → E → as: (cid:16) ( I dbl ) (cid:116)(cid:116) (cid:17) : (cid:16) ( L dbl ) (cid:116)(cid:116) (cid:17) → E → . Essentially, objects of (cid:32)L (cid:116) are pairs ( p : Y → X, φ ) where p is an arrow in E and φ isa right lifting structure with respect to the arrows in the image of I , but satisfyingonly the conditions (i) and (ii) since there is no non-trivial vertical composition. For a functor J : R → E → , there is similarly a category (cid:116) J : (cid:116) R → E → of left liftingstructures satisfying only conditions (i) and (ii).Note that in this construction, we have explicitly forgotten composition of liftingstructures: even for a mere functor I : L → E → , the category of right-lifting structures( I dbl ) (cid:116)(cid:116) : ( L dbl ) (cid:116)(cid:116) → Sq ( E )is a double category with non-trivial vertical composition.To conclude this section, it may be worth noting that taking double categories ofleft and right lifting structures are functorial constructions, and fact adjoint ones:( DBL / Sq ( E )) op DBL / Sq ( E ) ⊥ (cid:116)(cid:116) ( − )( − ) (cid:116)(cid:116) for which see Proposition 18 of Bourke and Garner [BG16]. It follows from Proposition2.11 that in that case also ( Cat / E → ) op Cat / E →⊥ (cid:116) ( − )( − ) (cid:116) is an adjunction. In this paper, we make extensive use of the notions of left and rightlifting structures in the context of algebraic weak factorization systems, which aredefined next.2.3. Algebraic weak factorisation systems.
In this section we recall the basicsof the theory of algebraic weak factorisation systems (AWFSs). Our main referenceis the paper by Bourke and Garner [BG16], which is consecrated solely to this topicand contains the most important results off-the-shelf. Another important sourceis Riehl [Rie11], who addresses some aspects in more depth, such as the subtletiesaround the distributive law. For the present purposes, we have already given themost important definitions in the previous section, namely that of a (left/right) lifingstructure with respect to a double category. It is this structure that enables us towork constructively in this paper, which is the main goal. In the present section, thisis combined with a functorial factorization.
Definition 2.13. A functorial factorization for a category E is a section of thecomposition functor E → dom × cod E → → E → . Spelling this out, it consists of a triple of functors
L, R : E → → E → , E : E → → E ,subject to two conditions. First, when f, f (cid:48) , h, k are morphisms in C with f (cid:48) .h = k.f ,the following diagram commutes:(2) A Ef BA (cid:48) Ef (cid:48) B (cid:48) h kLfLf (cid:48) RfRf (cid:48) E ( h, k )FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 23 Second, the top and bottom composites should compose to f , f (cid:48) . This decompositionyields natural transformations η : 1 ⇒ R , (cid:15) : L ⇒ f ) by the commutativesquares:(3) A EfB B η ⇒ Lf Rff (4)
A AEf B (cid:15) ⇒ Lf fRf An algebraic weak factorization system (AWFS) can be informally described as afunctorial factorization into a composite of two kinds of arrows, as follows. First,there are double categories I : L → Sq ( E ), J : R → Sq ( E ) induced by each other inthe following way: (cid:116)(cid:116) J ∼ = I and I (cid:116)(cid:116) ∼ = J in DBL / Sq ( E ) . Note that also (cid:116)(cid:116) (cid:0) I (cid:116)(cid:116) (cid:1) ∼ = I and (cid:0) (cid:116)(cid:116) J (cid:1) (cid:116)(cid:116) ∼ = J . Second, it is required that the functor L of the factorization factors through the vertical part I : L → E → (see Proposition2.11) of I , and that R factors through the vertical part J : R → E → .Taking the above as starting point, we can motivate the formal definition of anAWFS a bit better. First, observe that if the above holds, the lifting structure φ onthe arrows of the form Lf induces a family of maps δ f = φ Ef ,LLf ( RLf ):(5)
A ELfEf Ef Ef LLfLf RLfδ f Since L factors through L and R factors through R , this family defines a naturaltransformation: δ : L ⇒ LL.
Note that commutativity of the bottom triangle in (5) implies a counit law: (cid:15) Lf .δ f = 1 Lf . This counit law expresses that Lf is a coalgebra for the ‘mere co-pointed endofuntor’( L, (cid:15) ). Similarly, there is a family of maps µ f : RRf → Rf :(6) Ef EfERf B Ef RRfLRf Rfµ f which yields a natural transformation µ : RR ⇒ R which satisfies a unit law: µ f .η Rf = 1 Rf . Again, this law just expresses that Rf is an algebra for the ‘mere pointed endofunctor’( R, η ). Next, there is for any vertical arrow v in the double category R a filler for thediagram: A AEI ( v ) B A RI ( v ) LI ( v ) I ( v ) α since L factors through I : L → E → . Because the top triangle commutes, this fillerdefines an algebra structure α : RI ( v ) → I ( v ) for the pointed endofunctor R : E → →E → , η : 1 ⇒ R . Similarly, a vertical arrow u in the category L induces a coalgebrastructure β : I ( u ) → LI ( u ) for the co-pointed endofunctor L : E → → E → , (cid:15) : L ⇒ A BC D fgI ( u ) J ( v ) a filler can be given as a composite:(7) B BA EI ( u ) EI ( v ) DC C LI ( u ) fRI ( u )1 C E ( f, g ) gLI ( v ) RI ( v )1 B β αI ( u ) I ( v ) . We now turn to the definition of an algebraic weak factorisation system (AWFS).A consequence of this definition is that the lifts for ‘canonical’ diagrams, such as (6)and (5) or more generally for any L -coalgebra and R -algebra, agree with the generalshape (7). Definition 2.14.
Suppose (
L, R, (cid:15), η ) is a functorial factorization on a category E and δ : L ⇒ LL , µ : RR ⇒ R are natural transformations given by diagonals as indiagrams (5) and (6). Then ( L, R, δ, µ, (cid:15), η ) is an algebraic weak factorization system (AWFS) when the following conditions hold:(i) The triple (
L, δ, (cid:15) ) satisfies the conditions for a comonad structure on E (ii) The triple ( R, µ, η ) satisfies the conditions for a monad structure on E FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 25
Distributive laws between monads were introduced by Beck in [Bec69] as nat-ural transformations λ : T S ⇒ ST for monads S , T on some category, subjectto certain conditions relating to the monad structures of S and T . These areequivalent to a monad structure on the composition T S , subject to compati-bility conditions with S and T . As remarked by Power and Watanabe [PW02],there are many more distributive laws to consider, including one for a comonadover a monad, or the other way around. These are all different choices. Whatis meant by a distributive law for a comonad L over a monad R in this paperis the dual of definition 6.1 in [PW02]. Spelled out, this is a natural transfor-mation λ : LR ⇒ RL subject to the following conditions:(9) Rδ.λ = λL.Lλ.δ R , R(cid:15).λ = (cid:15) R µ L .Rλ.λR = λ.Lµ , η L = λ.Lη. Box 1.1.
Distributive laws combining monads and comonads(iii) The commutative square(8)
Ef ELfERf Ef δ f µ f LRf RLf whose diagonal is the identity 1 Ef : Ef → Ef , constitutes a distributive law LR ⇒ RL of L over R . We have put some background on this in Box 1.1, butwe will be mostly concerned with the alternative formulation of this conditionformulated in Proposition 2.15. The significance of the distributive law for anAWFS is pointed out in Theorem 2.21 below. The fact that it constitutes a‘distributive law’ in some broader sense will also become apparent from theAWFS we construct in this paper.The definition of an AWFS as given above might seem to demand an overwhelmingamount of additional structure on a functorial factorisation together with the naturaltransformations δ, µ . The following proposition shows that in fact, the definition ofan AWFS contains some redundancy and can be reduced to a couple of equationalidentities. For instance, the distributive law can be expressed using a single equationalidentity which combines δ and µ . This observation is due to Richard Garner, andsince it will be quite important for our subsequent verifications, we will refer to thisequation as the Garner equation . Proposition 2.15.
Suppose ( L, R, (cid:15), η ) is a functorial factorisation, and suppose δ : L ⇒ LL is a natural transformation over dom: E → E , and µ : RR ⇒ R is a naturaltransformation over cod: E → E . Then the following statements are equivalent:(i) The triples ( L, δ, (cid:15) ) , ( R, µ, η ) satisfy the conditions for a comonad over dom: E →E and a monad over cod: E → E respectively. i.e. the following equations aresatisfied: (10) RLf.δ f = 1 µ f .LRf = 1 E (1 , Rf ) .δ f = 1 µ f .E ( Lf,
1) = 1 δ Lf .δ f = E (1 , δ f ) .δ f µ f .µ Rf = µ f .E ( µ f , And the
Garner equation holds: (11) δ f .µ f = µ Lf .E ( δ f , µ f ) .δ Rf (ii) The diagram (8) commutes, its diagonal is the identity, and constitutes a dis-tributive law for L over R , in the sense that the equations (9) are satisfied.(iii) The tuple ( L, R, δ, µ, (cid:15), η ) is an AWFS.Proof. The only thing to prove is the equivalence of (i) and (ii), since these twocomplement each other to an AWFS. We leave it to the reader to spell out theaxioms (9) for λ =(8) and conclude that these contain precisely the counit, unit,coassociativity and associativity conditions as well as the Garner equation. (cid:3) In the next section, we will summarise the relationship between the above definitionof an AWFS and the informal description given earlier.2.4.
A double category of coalgebras.
In this section we assume that(
L, R, (cid:15), η, δ, µ )is an AWFS on a category E as in Definition 2.14. We would like to define a doublecategory L - C oalg whose objects are objects of E , whose vertical arrows are coalge-bras for the comonad ( L, δ, (cid:15) ) and whose squares are morphisms of algebras. For thisto make sense, it is needed to define a vertical composition of coalgebras.We can regard R and L as either mere (co)pointed endofunctors, or (co)monads. Inboth cases, (co)algebras for them can be represented as diagonal fillers for diagramsin E . Indeed, recall that a coalgebra β : f → Lf for the mere co-pointed endofunctor L is the same thing as a filler for the square: A E f B B f Lf B Rfβ These last two conditions mean that δ , µ are given by arrows δ f , µ f as above such that δ f .Lf = LLf and
Rf.µ f = RRf
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 27
Indeed, one can check that the co-unit condition dictates that the top arrow of thesquare β : f → Lf must be the identity, hence the top triangle commutes, and that R f .β = 1 B , hence the bottom triangle commutes. When we are interested in coalge-bras for the comonad ( L, δ, (cid:15) ), this arrow is subject to the co-associativity condition δ f .β = L ( β ) .β . This boils down to the equational identity: δ f .β = E (1 A , β ) .β (12)Similarly, an algebra structure Rf → f (for either the mere pointed endofunctoror the monad) is defined entirely by an underlying arrow β : E f → dom f . We willhence refer to algebra or coalgebra structures by their underlying map. For thecomonad L and the monad R , we denote the category of (co)algebras and morphismsof (co)algebras by L - Coalg and R - Alg respectively. As we have seen in (7), every L -coalgebra bears a left lifing structure with respect to every R -algebra. This takesthe form of a functor Φ: L - Coalg → (cid:116) R - Alg for which we can describe the image in the form of an extra condition on liftingstructures in the following proposition. This proposition is the dual of Lemma 1 inBourke-Garner [BG16].
Proposition 2.16.
The functor Φ: L - Coalg → (cid:116) R - Alg is injective on objects andfully faithful, and its image consists of those arrows ( i, φ ) with a left lifting structure φ for which the following diagrams commute: (13) A Eh ELhEhB C C i uv C δ h Rh RLhRh φ u , v ( R h ) v (cid:48) : = φ u , v ( R h ) φ δ h . u , v (cid:48) ( R L h ) Proof.
See Lemma 1 of Bourke-Garner [BG16]. The proof relies on the distributivelaw, or more precisely, the Garner Equation. (cid:3)
Corollary 2.17.
Suppose we have a pushout square:
A A (cid:48)
B B (cid:48) f f (cid:48) and suppose that β is a coalgebra structure on f . Then there is a unique coalgebrastructure β (cid:48) on f (cid:48) which makes the diagram into a morphism of coalgebras. Hence there is a cofibred structure (see the remarks above Definition 2.10) on E : (14) L - coalg : E → cocart → S ets which associates to every morphism the set of coalgebra structures on it.Proof. There is a unique left lifting structure with respect to R - Alg on f (cid:48) such thatthe square is a morphism of lifting structures. It is easy to see that this lifting struc-ture satisfies the condition of Proposition 2.16. Therefore it also defines a coalgebrastructure on f (cid:48) in such a way that the diagram is a morphism of coalgebras. Thisdefines the functor (14) on morphisms. (cid:3) With the construction introduced in Proposition 2.11, we can improve a bit on theabove Proposition, by regarding Φ as a double functor(15) Φ: L - Coalg dbl → (cid:116)(cid:116) R - Alg dbl . which is injective and fully faithful on vertical morphisms. One can check that thecondition on left lifting structures of Proposition 2.16 is closed under composition ofleft lefting structures. So we can inherit vertical composition from this category, anddefine a double category of coalgebras as follows: Definition 2.18.
Define the double category L - C oalg as the double image of thefunctor (15). This is called the double category of coalgebras for the algebraic weakfactorisation system ( L, R, (cid:15), η, δ, µ ). Similarly, the double category R - A lg of algebrasis defined as the double image of the transpose (cid:101) Φ: R - Alg dbl → L - Coalg dbl (cid:116)(cid:116) . as per Proposition 2.11.It will be useful for the rest of this paper to record an expression for the verticalcomposition of coalgebras induced by the previous definition. For instance, it will beused in Corollary 4.6 below to show that vertical composition of hyperdeformationretracts (HDRs) is the same as vertical composition of coalgebras for a certain AWFS.Also, we can use it to show where the distributive law for AWFSs is actually used inthe theory of Bourke and Garner [BG16].So suppose that f : A → B , g : B → C come with coalgebra structures β : f → Lf , γ : g → Lg , and write h : = g.f . By (7), both f and g have the left-lifting property withrespect to Rh – and indeed it turns out that their composition coalgebra structure isgiven by first lifting with respect to f and then g according to this recipe. Spelling FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 29 out (7), this filler is the diagonal in the below diagram:(16)
A Eh EhB Ef ERhEg ERhC C C Lh Eh LRh ERh
LfβLgfg E ( Lh, g ) µ h C RgE ( µ h .E ( Lh, g ) .β, C ) γ µ h RRh C Rh . Hence the candidate coalgebra structure for the composite h = g.f is given by κ : = µ h .E ( µ h .E ( Lh, g ) .β, C ) .γ : h → Lh (17) = µ h .E ( E (1 A , g ) .β, C ) .γ where the latter condition follows from one the unit law for µ . The fact that thiscandidate is a coalgebra structure is a result that follows from Proposition 2.16.Similarly, if f , g are algebras with algebra structures β , γ , their composition h : = g.f has algebra structure:(18) β.E (1 A , γ.E ( f, Rh ) .δ h ) .δ h = β.E (1 A , γ.E ( f, C )) .δ h Lemma 2.19.
Suppose ( L, R, (cid:15), η ) is a functorial factorization and δ : L ⇒ LL , µ : RR ⇒ R are natural transformations which make the diagrams (5) and (6) com-mute, i.e. with δ f , every arrow Lf is a coalgebra for the mere co-pointed endofunctor ( L, (cid:15) ) and with µ f , every Rf is an algebra for the mere pointed endofunctor ( R, η ) .Then for every h : A → C , the composition Rh.RLh has an algebra structure forthe pointed endofunctor ( R, η ) given by (18) , or explicitly: κ : = µ Lh .E (1 ELh , µ h .E ( RLh, C )) .δ Rh.RLh : E Rh.RLh → E Lh Further, the
Garner equation (11) is satified if and only if the following square isa morphism of algebras for the given structures:
Eh ELhEhC C C δ h Rh RLhRh
Proof.
The first claim follows from dualizing the preceding discussion on the com-position of coalgebras for the mere co-pointed endofunctors – the filler is again thediagonal in the dual counterpart of (16), which is (18).
So we focus on the second claim. As one can readily check, this comes down tothe identity:(19) κ.E ( δ h , C ) = δ h .µ h For this we have: κ.E ( δ h , C ) = µ Lh .E (1 ELh , µ h .E ( RLh, C )) .δ Rh.RLh .E ( δ h , C ) = µ Lh .E (1 ELh , µ h .E ( RLh, C )) .E ( δ h , E ( δ h , C )) .δ Rh = µ Lh .E ( δ h , µ h .E ( RLh.δ h , C )) .δ Rh = µ Lh .E ( δ h , µ h ) .δ Rh Where we used the identities E ( u, v ) .E ( s, t ) = E ( u.s, v.t )throughout, naturality of δ at = and at the last step the identity RLh.δ h = 1.Hence (19) precisely states the distributive law. (cid:3) From the above definitions, observe that(20) (cid:116)(cid:116) R - A lg → (cid:116)(cid:116) ( R - Alg dbl )is an inclusion and that Φ factors through it via the transpose of R - A lg (cid:44) → ( L - Coalg dbl ) (cid:116)(cid:116) . We have the following proposition:
Lemma 2.20.
Under the distributive law, the image of a vertical morphism in (cid:116)(cid:116) R - A lg along the functor (20) satisfies the condition of Proposition 2.16.Proof. It is easy to see that when ( δ h , C ) is an algebra morphism between Rh andthe composed algebra, then the property of Proposition 2.16 holds. So the statementfollows from Lemma 2.19. (cid:3) Since (cid:116)(cid:116) R - A lg can also be taken to define a cofibred structure (cid:116)(cid:116) R - A lg : E → cocart → S ets ,the previous proposition can be rewritten as a theorem on cofibred structures: Theorem 2.21.
Suppose ( L, R, (cid:15), η, δ, µ ) is an AWFS (Definition 2.14). Then:(i) The natural transformation between cofibred structures ϕ : L - coalg → (cid:116)(cid:116) R - A lg induced by Φ is an isomorphism(ii) The natural transformation between fibred structures (cid:101) ϕ : R - alg → L - C oalg (cid:116)(cid:116) induced by (cid:101) Φ is an isomorphism. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 31
Proof.
The two statements are dual. To prove (i), it is enough to show that thatfor each morphism v in E , ϕ v is an isomorphism, i.e. that coalgebra structures on v correspond precisely to left lifting structures on v with respect to R - A lg . Thisfollows from Lemma 2.20 and Proposition 2.16. (cid:3) As another consequence of the above theorem, the double functor L - Coalg dbl → (cid:116)(cid:116) R - A lg through which Φ factors is surjective and full on vertical morphisms and squaresapart from being injective and fully faithful on horizontal morphisms. Hence this,and similarly dual reasoning, induces equivalences of images: L - C oalg ∼ = (cid:116)(cid:116) R - A lg and R - A lg ∼ = L - C oalg (cid:116)(cid:116) (over Sq ( E ) )which is the desired property of an AWFS. To summarise, we have found two doublecategories over Sq ( E ) which are the categories of left or right lifting structures withrespect to each other. These are the double categories of coalgebras and algebras forthe AWFS, respectively.Before, moving on, we address the natural question of whether there could be adifferent vertical composition of algebras or coalgebras than the one used above. Asshown by Bourke and Garner in Proposition 4 of [BG16], this is not the case, inthe the sense that vertical composition of algebras, for a certain given monad R ,completely determines an AWFS which induces that composition: Proposition 2.22.
Suppose R : E → E is a monad over cod: E → E . Then thereis a bijection between extensions of R to an AWFS and extensions of R - Alg → E to a double category over Sq ( E ) . Under this bijection, the vertical composition ofalgebras coincides with the vertical composition induced by the AWFS.Proof. See [BG16], Proposition 4. The idea is that the unit of R determines L , and δ is determined by the unique morphism of algebras Rf → Rf.RLf induced from(
LLf, f → Rf.RLf , since Rf has the free algebra structure on f . (cid:3) Cofibrant generation by a double category.
Later, we will need to comparedifferent algebraic weak factorisation systems, or prove that they are the same. Away to do this is to look at generating double categories. The following is fromBourke-Garner [BG16]:
Definition 2.23.
Suppose J : J → Sq ( E ) is a double functor for a small doublecategory J . An AWFS is cofibrantly generated by J if J (cid:116)(cid:116) ∼ = R - A lg over Sq ( E ).When J is large, Bourke and Garner call this property class-cofibrantly generated .The dual property, when (cid:116)(cid:116) I ∼ = L - C oalg , is called (class)-fibrantly generated . Theconclusion reached in the previous section can be summarised as: Corollary 2.24 ([BG16], Proposition 20) . An AWFS is class-cofibrantly generated byits double category of coalgebras, and class-fibrantly generated by its double categoryof algebras.
We refer to Bourke-Garner to results of the type that say that under appropriateconditions (i.e. E locally presentable), the AWFS generated by any small doublecategory J → Sq ( E ) exists ([BG16], Proposition 23). These results rely on somekind of small object argument [Gar09]. We will not rely on these results for ourconstruction of an AWFS, since we work constructively from the outset. But they canbe useful for boiling down a constructive theory to a classical theory for comparison. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 33 Dominances
Algebraic weak factorisation systems from dominances.
In this sectionwe show how dominances give rise to algebraic weak factorisation systems. Propo-sition Proposition 3.2 can also be found in Bourke and Garner [BG16]. The rest ofthe section studies the (double) category of (co)algebras for the AWFS a bit moreclosely and in terms of (co)fibred structure. Throughout this section, E is a categorysatisfying the conditions stated at the beginning of Section 2. Definition 3.1.
A class of monomorphisms Σ in E is called a dominance on E if(1) every isomorphism belongs to Σ and Σ is closed under composition.(2) every pullback of a map in Σ again belongs to Σ.(3) the category Σ cart of morphisms in Σ and pullback squares between them hasa terminal object.Since taking the domain of a monomorphism in Σ has a left adjoint id: E → Σ cart ,sending an object to the identity on it, the domain of the terminal object in Σ cart isthe terminal object in E . We will denote this map by (cid:62) : 1 → Σ. Proposition 3.2.
Let Σ be a dominance. Then the functorial factorization given by B Lf (cid:47) (cid:47) M f = Σ a ∈ A Σ σ ∈ Σ B σa Rf (cid:47) (cid:47) A, with Lf ( b ) = ( f ( b ) , (cid:62) , λx.b ) and Rf ( a, σ, τ ) = a can be extended to an algebraic weakfactorization system.Proof. Note that
M f classifies Σ-partial maps into B over A . Let us spell out whatthis means. By a Σ-partial map X (cid:42) B over A we mean a pair consisting of a subob-ject m : X (cid:48) → X with m ∈ Σ (which does not depend on the choice of representative)and a map n : X (cid:48) → B making X (cid:48) n (cid:47) (cid:47) m (cid:15) (cid:15) B f (cid:15) (cid:15) X (cid:47) (cid:47) A commute. Note that such Σ-partial maps X (cid:42) B over A can be pulled back alongarbitrary maps Y → X . Saying that M f classifies Σ-partial maps into B over A means that any such map can be obtained by pulling back the Σ-partial map( Lf, B ): M f (cid:42) B along a unique map X → M f over A .For the monad structure, we need to define a map µ f making M Rf µ f (cid:47) (cid:47) RRf (cid:15) (cid:15) M fRf (cid:15) (cid:15) A (cid:47) (cid:47) A commute. Maps X → M Rf over A correspond to diagrams of the form X (cid:48)(cid:48) (cid:47) (cid:47) (cid:15) (cid:15) B f (cid:15) (cid:15) X (cid:48) (cid:15) (cid:15) X (cid:47) (cid:47) A with both inclusions X (cid:48)(cid:48) → X (cid:48) and X (cid:48) → X belonging to Σ. By considering thecomposition we get a map X → M f , naturally in X , so by Yoneda we obtain a map M Rf → M f as desired. The unit law and associativity are easily verified.For the comonad structure, we need to define a map δ f making B (cid:47) (cid:47) Lf (cid:15) (cid:15) B LLf (cid:15) (cid:15) M f δ f (cid:47) (cid:47) M Lf commute. Note that M Lf = (cid:88) ( a,σ,τ ) ∈ M f (cid:88) σ (cid:48) ∈ Σ ( B ( a,σ,τ ) ) σ (cid:48) . So if (( a, σ, τ ) , σ (cid:48) , τ (cid:48) ) ∈ M Lf and ∗ ∈ σ (cid:48) , then ( a, σ, τ ) = ( f ( b ) , (cid:62) , λx.b ) for b = τ (cid:48) ( ∗ );hence ∗ ∈ σ and τ ( ∗ ) = τ (cid:48) ( ∗ ). In other words, M Lf = { (( a ∈ A, σ ∈ Σ , τ ∈ ( B a ) σ ) , σ (cid:48) ∈ Σ , τ (cid:48) ∈ ( B a ) σ ) : σ (cid:48) ≤ σ, τ (cid:22) σ (cid:48) = τ (cid:48) } . So we can define a map δ f : M f → M Lf by sending ( a, σ, τ ) to (( a, σ, τ ) , σ, τ ). Counitlaws and coassociativity are easily verified.The distributive law (Garner equation) is a bit annoying and left to the reader. (cid:3) Proposition 3.3.
A coalgebra structure for f : B → A is unique, and it exists if andonly if f belongs to Σ .Hence, there is a cofibred structure (21) σ : E cocart → S ets where σ ( f ) contains a single element when f ∈ Σ , and is empty otherwise. Moreover,there is an isomorphism of cofibred structures between σ and the cofibred structure ofcoalgebras.Proof. We show that every f : B → A can be equipped with a coalgebra structure forthe copointed endofunctor M if and only if it belongs to Σ, and that the coalgebrastructure is indeed unique and always satisfies the coassociativity condition. Fromthis, it is easy to derive an isomorphism of cofibred structures in light of (21). FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 35
Suppose γ : A → M f is a map exhibiting f as a coalgebra for the copointed endo-functor M . In other words, we have Rf.γ = 1 A and γ makes B f (cid:15) (cid:15) (cid:47) (cid:47) B Lf (cid:15) (cid:15) A γ (cid:47) (cid:47) M f commute. These data correspond to a Σ-partial map A (cid:42) B : A (cid:48) m (cid:47) (cid:47) s (cid:15) (cid:15) B f (cid:15) (cid:15) A (cid:47) (cid:47) A where s ∈ Σ. Further, s fits in a pullback square, namely the middle square of thefollowing diagram: B A (cid:48) B A A M f Σ f n s L f mγ whence we find n : B → A (cid:48) such that m.n = 1. Because s.n.m = f.m = s and s ismonic, we also have n.m = 1 A (cid:48) . So A (cid:48) and B are isomorphic over A . It follows f ∈ Σand that γ classifies the map ( f, B ). From this it is clear that γ must be uniquewhenever it exists and that it will always satisfy the coassociativity condition. It alsofollows, incidentally, that the square at the beginning of the proof is a pullback.Conversely, if f ∈ Σ we can choose s = f and m = 1 and this gives us the coalgebrastructure for the copointed endofunctor we want. The second part of the propositionfollows immediately. (cid:3) Lastly, we briefly stop at algebras for the monad. From Theorem 2.21, we knowthat the fibred structure of algebras is isomorphic to the fibred structure of rightlifting structures with respect to the double category of coalgebras. It remains tocharacterize this category. To that end, we use the following:
Lemma 3.4. If f : B → A and f (cid:48) : B (cid:48) → A (cid:48) are coalgebras, then a pair of maps ( m : B (cid:48) → B, n : A (cid:48) → A ) making B (cid:48) f (cid:48) (cid:15) (cid:15) m (cid:47) (cid:47) B f (cid:15) (cid:15) A (cid:48) n (cid:47) (cid:47) A commute is a morphism of coalgebras if and only if the square is a pullback. Proof.
It is not hard to check that such a pullback square constitutes a morphism ofcoalgebras.For the converse, let us first make the following observation. Suppose γ : A → M f is a coalgebra structure on f . Then γ fits into a diagram of the form B f (cid:15) (cid:15) (cid:47) (cid:47) B Lf (cid:15) (cid:15) (cid:47) (cid:47) (cid:62) (cid:15) (cid:15) A γ (cid:47) (cid:47) M f (cid:47) (cid:47) Σwhere M f → Σ is the obvious projection. Note that the right hand square is alwaysa pullback and that the left hand square is as well, as we saw in the previous proof.So the outer rectangle is a pullback.So if B (cid:48) f (cid:48) (cid:15) (cid:15) m (cid:47) (cid:47) B f (cid:15) (cid:15) A (cid:48) n (cid:47) (cid:47) A is a morphism of coalgebras, then this fits into a commutative diagram of the form B (cid:48) f (cid:48) (cid:15) (cid:15) m (cid:47) (cid:47) B f (cid:15) (cid:15) (cid:47) (cid:47) (cid:62) (cid:15) (cid:15) A (cid:48) n (cid:47) (cid:47) A (cid:47) (cid:47) Σin which the right hand square and the outer rectangle are pullbacks. Therefore theleft hand square is a pullback as well. (cid:3)
Hence we can deduce:
Corollary 3.5.
Let Σ be the double category whose horizontal arrows are arbitraryarrows from E , whose vertical arrows are maps from Σ and whose squares are pullbackssquares. Then:(i) There is an isomorphism between the following notions of fibred structure: • Algebras for the AWFS induced by Σ ; • Right lifting structures with respect to Σ .(ii) There is a functor R - Alg → Σ (cid:116) given on objects by (i) which is fully faithful.(iii) There is an equivalence of double categories R - A lg ∼ = Σ (cid:116)(cid:116) whose vertical restriction is prescribed by (ii). FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 37
Proof.
Because Σ is, essentially, the double category of coalgebras for the comonad. (cid:3) Definition 3.6.
The algebras or right lifting structure of Corollary 3.5 are called trivial fibrations . The fibred structure (i), the category (ii), and the double category(iii) are respectively denoted by trivFib , TrivFib , and T rivFib . AWFS from Moore structure
A category with Moore structure.
In this section we construct an algebraicweak factorisation system on a category with Moore structure E . We will then studyits coalgebras and algebras more closely. We show that the structure of a coalgebrais equivalent to the structure of a hyperdeformation retraction (HDR). Algebras, onthe other hand, will be identified as naive fibrations .Categories with Moore structure are a modification of the path object categories introduced by Van den Berg-Garner [BG12]. In that paper, a weak factorisationsystem is constructed for an arbitrary path object category. Here, we use a newmethod to show that it is also possible to construct an algebraic weak factorisationsystem. To that end, we do need to modify the axioms of a Moore structure relativeto the original definition – but note that ours is neither weaker nor stronger.A full definition of a Moore structure can be found in the Appendix, Definition A.1.The differences between the definition here and the one in [BG12] are the following:(1) The coassociativity condition: M Γ . Γ = Γ M. Γ: M X → M M X. which turns Γ into a comonad;(2) The distributive law:Γ .µ = µ. ( M µ.ν. (Γ .p , α. ( p , M ! .p )) , Γ .p ): M X × X M X → M M X. where we have defined the natural transformation α : X × M → X by α X : = θ MX .α MX, with θ, α − , − as in Remark A.3 in the Appendix. A diagrammaticillustration of this condition can be found in equation (36) in the same place.(3) The ‘twist map’ τ : M ⇒ M is dropped, only to return in Section 5.2 as symmetric Moore structure.In addition, we will also introduce a notion of two-sided
Moore structure in the latersections.4.2.
An algebraic weak factorisation system.
This subsection shows how everyMoore structure bears an algebraic weak factorisation system. A similar result, inabsence of distributive laws, can also be found in North ([Nor19], Theorem 3.28).Here, we improve on this by showing that our distributive law for Moore structuresimplies the distributive law for the Bourke-Garner definition of an AFWS. In theshort subsections that follow, we address each one of the requirements of Proposition2.15 (i).4.2.1.
Functorial factorisation.
First of all, if f : A → B is a morphism, we can factorit as A Lf (cid:47) (cid:47) Ef Rf (cid:47) (cid:47) B FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 39 by putting Ef = M B × B A (pullback of t and f ), Lf = ( r.f,
1) and Rf = s.p . Inthe obvious way E , L and R extend to functors, and the whole factorisation is readilyseen to be functorial.4.2.2. The comonad.
The comultiplication δ f needs to be a mapping filling: A Lf =( r.f, (cid:15) (cid:15) (cid:47) (cid:47) A LLf =( r. ( r.f, , (cid:15) (cid:15) M B × B A δ f (cid:47) (cid:47) M ( M B × B A ) × MB × A A where the object in the lower right-hand corner is the pullback of t MB × B A and Lf .Note that there is a mediating natural isomorphism( M M B × MB M A ) × MB × B A A ( ν, −−−→ M ( M B × B A ) × MB × B A A where ν is the natural isomorphism induced by pullback-preservation of M . Hencewe can put:(22) δ f = ( ν. (Γ .p , α. ( p , M ! .p )) , p )where θ denotes the natural isomorphism M p : M ( X × → M X . Curiously, δ f in no way refers to f . We can check it is well-defined. For the part mapping into M M B × MB M A , we have:
M t. Γ .p = α. ( t, M !) .p = α. ( t.p , M ! .p )= α. ( f.p , M ! .p )= M f.α. ( p , M ! .p ) . Further, t. Γ .p = r.t.p = r.f.p . In addition, the square commutes, as one easilychecks.From here it remains to check the comonad laws given in Proposition 2.15. Firstof all: s.p .δ f =( s. Γ .p , s.α. ( p , M ! .p ))=( p , p )=1and ( M ( s.p ) .p , p ) .δ f =( M s. Γ .p , p )=( p , p )=1 , and hence the counit laws are satisfied. Second, we check coassociativity: E (1 , δ f ) =( M δ f .p , p ) .δ f ==( ν. ( ν. ( M Γ .M p .p , M α.ν. ( M p .p , M M ! .M p .p )) , M p .p ) , p ) .δ f =( ν. ( ν. ( M Γ . Γ .p , M α. ( α. ( p , M ! .p ) , M M ! . Γ .p )) ,M α. ( p , M ! .p )) , p )= ( ν. ( ν. (Γ . Γ .p , Γ .α. ( p , M ! .p )) , α. ( p , M ! .p )) , p )= ( ν. (Γ .p , α. ( p , M ! .p )) , p ) .δ f = δ L f .δ f where we have used the strength of Γ with respect to α at = and the identity M ! .ν. (Γ .p , α. ( p , M ! .p )) = M ! .p at = .4.2.3. The monad.
The multiplication is given by:(23)
M B × ( t,s.p ) ( M B × B A ) M B × B AB B µ f s.p s.p with µ f = ( µ. ( p , p .p ) , p .p ), reminding the reader that we write the arguments to µ in sequential order rather than order of categorical composition ( µ is to be thoughtof as path concatenation). Also note that again, the definition of the multiplicationformula in no way refers to f . The monad laws are easy given that M B is an internalcategory.4.2.4.
The distributive law.
We should check the Garner equation (11), given by theidentity δ f .µ f = µ Lf .E ( δ f , µ f ) .δ Rf (24)for every f . We unfold (24) for the definitions presented above. We do so by (asbefore) explicitly by writing ν for the (family of) mediating isomorphisms:(25) ν : M A × MC M B → M ( A × C B )For this, the expression for the left-hand side of (24) amounts to: δ f .µ f =( ν. (Γ .p , α. ( p , M ! .p )) , p ) . ( µ. ( p , p .p ) , p .p ) =( ν. (Γ .µ. ( p , p .p ) , α. ( p .p , M ! .µ. ( p .p , p ))) , p .p ) FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 41
The right-hand side amounts to: µ Lf .E ( δ f , µ f ) .δ Rf =( µ. ( p , p .p ) , p .p ) . ( M ( µ. ( p , p .p ) , p .p ) .p , ( ν. (Γ .p , α. ( p , M ! .p )) , p ) .p ) . ( ν. (Γ .p , α. ( p , M ! .p )) , p ) =( µ. ( p , p .p ) , p .p ) . ( M ( µ. ( p , p .p ) , p .p ) .ν. (Γ .p , α. ( p , M ! .p )) , ( ν. (Γ .p , α. ( p , M ! .p )) , p ) .p ) =( µ. ( M ( µ. ( p , p .p ) , p .p ) .ν. (Γ .p , α. ( p , M ! .p )) ,p . ( ν. (Γ .p , α. ( p , M ! .p )) , p ) .p ) , p .p )Since the second components of the two expressions are the same, we can focus onthe first component. We proceed by reducing this first component for the right-handside as follows: µ. ( M ( µ. ( p , p .p ) , p .p ) .ν. (Γ .p , α. ( p , M ! .p )) ,p . ( ν. (Γ .p , α. ( p , M ! .p )) , p ) .p ) = µ. ( M ( µ. ( p , p .p ) , p .p ) .ν. (Γ .p , α. ( p , M ! .p )) ,ν. (Γ .p , α. ( p , M ! .p )) .p ) = µ. ( ν. ( M µ.ν. ( p , M ( p ) .p ) , M ( p ) .p ) . (Γ .p , α. ( p , M ! .p )) ,ν. (Γ .p , α. ( p , M ! .p )) .p ) = µ. ( ν. ( M µ.ν. (Γ .p , M ( p ) .α. ( p , M ! .p )) , M ( p ) .α. ( p , M ! .p )) ,ν. (Γ .p , α. ( p , M ! .p )) .p ) = µ. ( ν. ( M µ.ν. (Γ .p , α. ( p .p , M ! .p )) , α. ( p .p , M ! .p )) ,ν. (Γ .p .p , α. ( p .p , M ! .p .p ))) = ν. ( µ. ( M µ.ν. (Γ .p , α. ( p .p , M ! .p )) , Γ .p .p ) ,µ. ( α. ( p .p , M ! .p ) , α. ( p .p , M ! .p .p ))) = ν. (Γ .µ. ( p , p .p ) , α. ( p .p , M ! .µ. ( p .p , p )))Here = uses the identity ν. ( M µ. ( p , M p .p ) , M p .p ) = M ( µ. ( p , p .p ) , p .p ) .ν which is just naturality of ν . At = we have twice used naturality of α and absorbeda projection term in the right term of the composite. At that point, we are left witha nice expression of the form: µ.ν × ν : ( M A × MB M M B ) × A × B MB ( M A × MB M M B ) → ( M A × MB M M B )Naturality of µ and ν implies that this map can be rewritten as(26) ν. ( µ. ( p .p , p .p ) , µ. ( p .p , p .p ))which we have done at = . In the resulting expression, we recognize in the rightcomponent the composition of two constant paths, which is the constant path on the composition, and in the left component the distributed term of the distributive lawbetween µ and Γ, which yields the desired equality.We can summarise the result of this section as follows: Proposition 4.1.
Suppose E is a category with Moore structure. Then the functorialfactorisation ( L, R, (cid:15), η ) given by A M B × B A B Lf : = ( r.f, Rf : = s.p together with the natural transformations δ : L ⇒ LL defined by (22) and µ : RR ⇒ R defined by (23) constitutes an algebraic weak factorisation system (AWFS) on E inthe sense of Definition 2.14. (cid:3) Coalgebras.Definition 4.2.
Let i : A → B be a map. To equip i with a hyperdeformation retrac-tion means specifying a map j : B → A and a homotopy H : B → M B such that thefollowing hold: j.i = 1 A , s.H = 1 B , t.H = i.j, M H.H = Γ .H. If such a structure can be specified for i , we will call it a hyperdeformation retract . Remark 4.3.
The maps t and Γ equip any r : X → M X with the structure of ahyperdeformation retraction.
Proposition 4.4.
The function which associates to every i : A → B the set { ( H, j ) | ( i, j, H ) is an HDR } can be extended to a presheaf on the category of arrows of E and cocartesian (pushout)squares: hdr : E → cocart → S ets. So HDRs form a cofibred structure on E .Proof. It is sufficient to show that HDRs are closed under pushouts in a way com-patible with composition of pushout squares. The reader is invited to do this as anexercise, as it will also be clear from the proof of Proposition 4.5. (cid:3)
Proposition 4.5.
The following notions of cofibred structure are isomorphic: • Having a coalgebra structure with respect to L = ( r.f, , • Having the structure of an HDR.Proof.
Suppose i : A → B is an arrow in E . The map i : A → B carries a coalgebrastructure if there is a map ( H, j ) making A (cid:47) (cid:47) i (cid:15) (cid:15) A (1 ,r.i ) (cid:15) (cid:15) B ( H,j ) (cid:47) (cid:47) M B × B A FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 43 commute (which means i.j = t.H, j.i = 1 , H.i = r.i ) and such that s.p . ( H, j ) = 1(that is, s.H = 1) and δ. ( H, j ) = ( p .M ( H, j ) , p ) . ( H, j ). The latter condition meansthat δ. ( H, j ) = ( ν. (Γ .p , α. ( p , M ! .p )) , p ) . ( H, j ) = ( ν. (Γ .H, α. ( j, M ! .H )) , j )should equal ( M ( H, j ) .p , p ) . ( H, j ) = ( ν. ( M H.H, M j.H ) , j )(using naturality of ν ). In other words, that α. ( j, M ! .H ) = M j.H and Γ .H = M H.H .To summarise, a coalgebra structure is a hyperdeformation retraction satisfying, addi-tionally,
H.i = r.i and M j.H = α. ( j, M ! .H ). We now show that these two conditionsare always satisfied.First of all, we have H.i = H.i.j.i = H.t.H.i = t.M H.H.i = t. Γ .H.i = r.t.H.i = r.i.j.i = r.i, showing that a hyperdeformation retraction is automatically a strong deformationretraction.Secondly, to show M j.H = α. ( j, M ! .H ), we calculate: M i.M j.H = M ( i.j ) .H = M ( t.H ) .H = M t.M H.H = M t. Γ .H = α. ( t, M !) .H = α. ( t.H, M ! .H )= α. ( i.j, M ! .H )= M i.α. ( j, M ! .H ) . Since
M i is monic (even split monic), this proves the claim. We leave it to the readerto verify that this construction is functorial and induces an isomorphism of cofibredstructures. (cid:3)
Corollary 4.6.
HDRs admit a vertical composition, given by vertical compositionof coalgebras for an AWFS. Explicitly, the composition of two HDRs i : A → B , i , B → C is given by i .i with inverse map j .j and deformation (27) H ∗ H : = µ. ( H , M i .H .j ) Proof.
We only need to verify that the given formula and inverse represents compo-sition of coalgebras, as defined by the formula (17) above. Expanding this formula for h : = i .i yields:( H , j ) ∗ ( H , j )= µ h .E ( E (1 A , i ) . ( H , j ) , C ) . ( H , j )= ( µ. ( p , p .p ) , p .p ) . ( M C .p , ( M i .p , p ) . ( H , j ) .p ) . ( H , j )= ( µ. ( p , p .p ) , p .p ) . ( H , ( M i .H , j ) .j )= ( µ. ( H , M i .H .j ) , j .j )whence the statement follows. (cid:3) Similarly, we have the rest of the structure of a double category, whose verticalmorphisms are HDRs. In the rest of this section, we will study this structure inmore depth and using HDRs rather than coalgebras. If ( i (cid:48) : A (cid:48) → B (cid:48) , j (cid:48) , H (cid:48) ) and( i : A → B, j, H ) are HDRs, then a morphism of HDRs is defined as a pair of maps f : A (cid:48) → A and g : B (cid:48) → B such that A (cid:48) i (cid:48) (cid:15) (cid:15) f (cid:47) (cid:47) A i (cid:15) (cid:15) B (cid:48) g (cid:47) (cid:47) j (cid:48) (cid:15) (cid:15) B j (cid:15) (cid:15) A (cid:48) f (cid:47) (cid:47) A commutes and M g.H (cid:48) = H.g : B (cid:48) → M B .4.3.1.
HDRs are comonadic.
Consider the codomain functor:(28) cod:
HDR → E . The following facts will be helpful in Proposition 4.10 below.
Lemma 4.7. If ( i, j, H ) is an HDR, then A i (cid:15) (cid:15) i (cid:47) (cid:47) B H (cid:15) (cid:15) B r (cid:47) (cid:47) M B is a pullback. In particular, i is the equalizer of the pair r, H : B → M B .Proof. If i : A → B is an HDR, as witnessed by j : B → A and H : B → M B , then A i (cid:15) (cid:15) i (cid:47) (cid:47) B H (cid:15) (cid:15) j (cid:47) (cid:47) A i (cid:15) (cid:15) B r (cid:47) (cid:47) M B t (cid:47) (cid:47) B FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 45 exhibits i as a retract of H . Since H is monic ( s.H = 1), Lemma 4.8 (see below) tellsus that the left hand square is a pullback. (cid:3) Lemma 4.8.
If the commutative diagram A (cid:47) (cid:47) m (cid:15) (cid:15) C (cid:47) (cid:47) n (cid:15) (cid:15) A m (cid:15) (cid:15) B (cid:47) (cid:47) D (cid:47) (cid:47) B exhibits m as a retract of n and n is a monomorphism, then the left hand square is apullback.Proof. This is dual to Lemma B.2 in the Appendix. A reference is given there. (cid:3)
Corollary 4.9.
Suppose ( i (cid:48) , j (cid:48) , H (cid:48) ) , ( i, j, H ) are HDRs and g : B (cid:48) → B satisfies M g.H (cid:48) = H.g . Then there is a unique f : A (cid:48) → A such that ( f, g ) is a morphismof HDRs.Proof. The unique f is defined by virtue of Lemma 4.7. It remains to verify thatsquare j (cid:48) → j between the retracts commutes: j.g = j.i.j.g = j.t.H.g = t.M j.H.g = t.M j.M g.H (cid:48) = t.M ( j.g ) .H (cid:48) = j.g.t.H (cid:48) = j.g.i (cid:48) .j (cid:48) = j.i.f.j (cid:48) = f.j (cid:48) So the conclusion follows. (cid:3)
The following proposition combines the previous two observations and will be putto use later:
Proposition 4.10.
The functor (28) is comonadic, where the corresponding comonadis just the Moore functor M : E → E with comonad structure ( s, Γ) . Specifically (29) cod: HDR → M - Coalg , which sends ( i, j, H ) to ( B, H : B → M B ) , is an equivalence of categories.Proof. By Definition 4.2, the functor (29) has the correct codomain, i.e. it maps into M -coalgebras. By Corollary 4.9, the functor is full and faithful. Further, Lemma 4.7and its proof imply that that it is essentially surjective. So we have an equivalenceof categories. (cid:3) We can use the previous result to prove the following:
Corollary 4.11.
The category of HDRs has pullbacks.Proof.
By Proposition 4.10, the functor (28) creates limits preserved by M . So itfollows because M preserves pullbacks. (cid:3) To end this section, we record the following fact about HDRs, which holds whenthe unit r X : X → M X is a cartesian natural transformation. Note this happens tobe true for simplicial sets – but it is not an assumption of our theory.
Proposition 4.12.
When r X : X → M X is a cartesian natural transformation, everymorphism of HDRs is a pullback square.Proof. If B (cid:48) i (cid:48) (cid:15) (cid:15) f (cid:47) (cid:47) B i (cid:15) (cid:15) A (cid:48) g (cid:47) (cid:47) A is (the top part of) a morphism of HDRs, then it fits into a commutative cube: B i (cid:15) (cid:15) i (cid:47) (cid:47) A H (cid:15) (cid:15) B (cid:48) i (cid:48) (cid:47) (cid:47) i (cid:48) (cid:15) (cid:15) f (cid:62) (cid:62) A (cid:48) g (cid:59) (cid:59) H (cid:48) (cid:15) (cid:15) A r (cid:47) (cid:47) M AA (cid:48) r (cid:47) (cid:47) g (cid:62) (cid:62) M A (cid:48) Mg (cid:59) (cid:59) In this cube front and back faces are pullbacks (by Lemma 4.7), as is the bottomface (because r is a cartesian natural transformation). Therefore the top face is apullback as well. (cid:3) HDRs are bifibred.
In this section, we will study the domain functor
HDR →E . Definition 4.13.
A morphism of HDRs will be called a cartesian morphism of HDRs if also the bottom part, i.e. the square for j (cid:48) and j , is a pullback. Definition 4.14.
A morphism of HDRs will be called a cocartesian morphism ofHDRs if the square for i (cid:48) , i and the square for j (cid:48) , j are pushouts.In the following proposition, we denote by HDR the vertical part, as in Proposition2.11, of the double category of HDRs, i.e. the category of HDRs and morphisms ofHDRs.
Proposition 4.15.
The domain functor dom:
HDR → E is a bifibration , i.e. a Grothendieck fibration as well as an opfibration, where cartesianmorphisms are given by Definition 4.13, and cocartesian morphisms are given by
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 47
The following definition of the
Beck-Chevalley condition for bifibrations origi-nates from B´enabou-Roubaud [BR70]. For a bifibration
F → E , the conditionis satisfied when, as drawn in the diagram below, for every commutative squarein the fibre above a pullback square(30) • •• •• •• • f (cid:48) g k (cid:48) fl (cid:48) k lg (cid:48) such that f (cid:48) is cartesian and g is cocartesian, one has that f is cartesian ifand only if g (cid:48) is cocartesian. Note that in fact it is equivalent that only one ofthese two directions hold.If the bifibration comes with a choice of cartesian and cocartesian lifts in theform of fibrewise pullback ( − ) ∗ and pushforward ( − ) ∗ , this can be written asan isomorphism: l (cid:48)∗ k (cid:48)∗ ∼ = k ∗ l ∗ for every pullback square in the base as drawn. Box 1.2.
The Beck-Chevalley condition for bifibrations
Definition 4.14. Moreover, this bifibration satisfies the Beck-Chevalley condition (seeBox 1.2).
Before proving the proposition, we prove the following two lemmas:
Lemma 4.16.
Suppose r is the universal retract, in other words the category • i (cid:47) (cid:47) • j (cid:47) (cid:47) • with j.i = 1 . Then the functor ev : E r → E which sends a retract pair ( i, j ) to dom i is a bifibration satisfying the Beck-Chevalleycondition. Proof.
To prove that it is a fibration, suppose i : A → B , j : B → A is a retract pairand suppose f : A (cid:48) → A is any morphism. Then we can form a double pullback A (cid:48) f (cid:47) (cid:47) i (cid:48) :=(1 ,i.f ) (cid:15) (cid:15) A i (cid:15) (cid:15) B (cid:48) g (cid:47) (cid:47) j (cid:48) (cid:15) (cid:15) B j (cid:15) (cid:15) A (cid:48) f (cid:47) (cid:47) A resulting in a morphism of retract pairs ( i (cid:48) , j (cid:48) ) → ( i, j ). It is enough to verify thatthis morphism is cartesian over f , which is very easy.Similarly, for a pair ( i (cid:48) : A (cid:48) → A, j (cid:48) : B (cid:48) → B ) and a morphism f : A (cid:48) → A , thedouble pushout diagram A (cid:48) f (cid:47) (cid:47) i (cid:48) (cid:15) (cid:15) A i (cid:15) (cid:15) B (cid:48) g (cid:47) (cid:47) j (cid:48) (cid:15) (cid:15) B [ f.j (cid:48) , (cid:15) (cid:15) A (cid:48) f (cid:47) (cid:47) A yields a retract pair ( i, [ f.j (cid:48) , f .For the Beck-Chevalley condition we prove the ‘ ⇒ ’ direction of the definition ex-plained in Box 1.2. So consider the situation of (30) for the case at hand, i.e. adiagram: A (cid:48) AB (cid:48) BA (cid:48) AC (cid:48) CD (cid:48) DC (cid:48) C f (cid:48) ff gg (cid:48) g where the bottom and top squares pullbacks, the back (vertical) squares form a doublepullback, and the right-hand vertical squares form a double pushout. It needs to beshown that if the front vertical squares are a double pullback, then the left verticalsquares are a double pushout. But this follows immediately from the assumption on E stated at the beginning of Section 2. (cid:3) FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 49
Lemma 4.17.
Suppose we have a morphism of retract pairs ( i (cid:48) , j (cid:48) ) → ( i, j ) given by f : A (cid:48) → A , g : B (cid:48) → B . Then if this morphism is(i) a cartesian morphism of retract pairs, and H : B → M B gives ( i, j ) the structureof an HDR, then there is a unique HDR structure on the pair ( i (cid:48) , j (cid:48) ) such thatthe cartesian morphism is a cartesian morphism of HDRs.(ii) a cocartesian morphism of retract pairs, and H : B (cid:48) → M B (cid:48) gives ( i (cid:48) , j (cid:48) ) thestructure of an HDR, then there is a unique HDR structure on the pair ( i, j ) such that the cocartesian morphism is a cocartesian morphism of HDRs.Proof. (i): Since any HDR structure H (cid:48) that would be a witness to the claim makes B (cid:48) H (cid:48) (cid:34) (cid:34) H.g (cid:37) (cid:37) α. ( j (cid:48) ,M ! .H (cid:48) ) (cid:30) (cid:30) M B (cid:48) Mj (cid:48) (cid:15) (cid:15) Mg (cid:47) (cid:47) M B Mj (cid:15) (cid:15) M A (cid:48) Mf (cid:47) (cid:47) M A commute (see the proof of Proposition 4.5), it must be unique because M preservespullback squares. It remains to see that H (cid:48) can be defined in this way by setting H (cid:48) = ( α. ( j (cid:48) , M ! .H.g ) , H.g ) . To check that Γ .H (cid:48) = M H (cid:48) .H (cid:48) , it suffices to prove that both sides become equalupon postcomposing with both M M g and
M M j (cid:48) . But we have:
M M g.M H (cid:48) .H (cid:48) = M ( M g.H (cid:48) ) .H (cid:48) = M ( H.g ) .H (cid:48) = M H.M g.H (cid:48) = M H.H.g = Γ .H.g = Γ .M g.H (cid:48) = M M g. Γ .H (cid:48) and M M j. Γ .H (cid:48) = Γ .M j.H (cid:48) = Γ .α ( j (cid:48) , M ! .H.g )= M α. ( α. ( j (cid:48) , M ! .H.g ) , Γ .M ! .H.g )= M α. ( M j.H (cid:48) , M M ! . Γ .H.g )= M α. ( M j.H (cid:48) , M M ! .M H.H.g )= M α. ( M j.H (cid:48) , M M ! .M H.M g.H (cid:48) )= M ( α ( j, M ! .H.g )) .H (cid:48) = M ( M g.H (cid:48) ) .H (cid:48) = M M g.M H (cid:48) .H (cid:48) . Note we have used an identity from the proof of Proposition 4.5 here. Lastly, weverify: t.H (cid:48) = t. ( α. ( j (cid:48) , M ! .H.g ) , H.g )= ( t.α. ( j (cid:48) , M ! .H.g ) , t.H.g )= ( j (cid:48) , i.j.g )= (1 , i.f ) .j (cid:48) (ii): This is Proposition 4.4. (cid:3) We can now prove the above stated proposition:
Proof of Proposition 4.15.
By the previous two lemmas, it remains to show that when( i , j , H ), ( i , j , H ) and ( i , j , H ) are HDRs such that we have a composite A i (cid:15) (cid:15) f (cid:48) (cid:47) (cid:47) A i (cid:15) (cid:15) f (cid:47) (cid:47) A i (cid:15) (cid:15) B j (cid:15) (cid:15) g (cid:48) (cid:47) (cid:47) B j (cid:15) (cid:15) g (cid:47) (cid:47) B j (cid:15) (cid:15) A f (cid:48) (cid:47) (cid:47) A f (cid:47) (cid:47) A which is a morphism of HDRs, then:(i) If the right morphism of retract pairs is a cartesian morphism of HDRs, the leftone is automatically a morphism of HDRs(ii) If the left morphism is a cocartesian morphism of HDRs, then the right one isautomatically a morphism of HDRs.(i) follows again by taking projections: M g.M g (cid:48) .H = M ( g.g (cid:48) ) .H = H . ( g.g (cid:48) ) = ( H .g ) .g (cid:48) = M g. ( H .g (cid:48) ) FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 51 and
M j .M g (cid:48) .H = M ( f (cid:48) .j ) .H = M ( f (cid:48) ) .α. ( j , M ! .H )= α. ( f (cid:48) .j , M ! .H )= α. ( f (cid:48) .j , M ! .H .g.g (cid:48) )= α. ( j , M ! .H .g ) .g (cid:48) = α. ( j , M ! .H ) .g (cid:48) = M j .H .g (cid:48) (ii) follows again from Proposition 4.5, since the property is easy to verify forcoalgebras using Proposition 2.16.Observe that the Beck-Chevalley condition is now inherited from Lemma 4.16. (cid:3) The following is a first consequence of the more abstract approach we have takenso far:
Corollary 4.18.
In the category of HDRs, the pullback of a cartesian morphism ofHDRs along a morphism of HDRs exists and is a cartesian morphism of HDRs.Proof.
This is a direct consequence of the fact that dom:
HDR → E is a Grothendieckfibration (and that E has pullbacks). (cid:3) Algebras.
Recall that the monad for the AWFS defined in this section is givenby Rp = s.p , so that algebras are fillers: Y YM X × X Y X ( r.p, ps.p β satisfying an additional unit and associativity condition. Like with coalgebras, wecan characterise these in a different way. We take the following definition: Definition 4.19.
Let p : Y → X be any map. To equip p with transport structure means specifying a map T : M X × X Y → Y where M X × X Y is the pullback of t and p , with p.T = s.p , T. ( r.p,
1) = 1 and suchthat T. ( µ. ( p .p , p .p ) , p ) = T. ( p .p , T. ( p .p , p )) : ( M X × X M X ) × X Y → Y. where the first pullback is the pullback of t and s (the domain of µ ) and the secondof t.p and p . Proposition 4.20.
The function which associates to every p : Y → X the set { T : M X × X Y → Y | T is a transport structure for p } can be extended to a fibred structure ( E → cart ) op → S ets Proof.
This is easy when considering that a transport structure amounts to the samething as an algebra, so we leave this as part of Proposition 4.21. (cid:3)
Proposition 4.21.
The following notions of fibred structure are isomorphic: • Transport structure, • The structure of an algebra with respect to R = s.p .Proof. This is straightforward by unfolding the definition of an algebra. (cid:3)
The following definition (and Proposition 4.24) characterises algebras in terms ofan internal notion of path lifting, in similar vein to what was established in Van denBerg-Garner [BG12].
Definition 4.22.
A map p : Y → X together with an arrow L : M X × X Y → M Y in E is said to be a naive fibration when it satisfies the conditions:(i) ( M p, t ) .L = 1;(ii) L. ( r.p,
1) = r ;(iii) L. ( µ. ( p .p , p .p ) , p ) = µ. ( L. ( p .p , s.L. ( p .p , p )) , L. ( p .p , p ));(iv) Γ .L = M L.p .δ p . Proposition 4.23.
The function which associates to every p : Y → X the set { L : M X × X Y → M Y | ( p, L ) is a naive fibration } can be extended to a fibred structure nFib : ( E → cart ) op → S ets Proof.
Again, we leave this as part of their characterisation in Proposition 4.24. (cid:3)
Proposition 4.24.
Let p : Y → X be a map. If L specifies a naive fibration structureon p , then T = s.L is a transport structure on p . And if T is a transport structure on p , then L = M T.p .δ p turns p into a naive fibration. These operations are mutuallyinverse and define an isomorphism between the following notions of fibred structure: • Transport structure, • Naive fibrations.
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 53
Proof.
Suppose L satisfies the conditions (i)-(iv) of Definition 4.22, and let T = s.L .Then p.T = p.s.L = s.M p.L = s.p , T. ( r.p,
1) = s.L. ( r.p,
1) = s.r = 1, and T. ( µ. ( p .p , p .p ) , p ) = s.L. ( µ. ( p .p , p .p ) , p )= s.µ. ( L. ( p .p , s.L. ( p .p , p )) , L. ( p .p , p ))= s.L. ( p .p , s.L. ( p .p , p ))= T. ( p .p , T. ( p .p , p )) , so T is a transport structure. In addition, M T.p .δ p = M s.M L.p .δ p = M s. Γ .L = L, so L can be reconstructed from T .Conversely, suppose T is a transport structure on p , and define L = M T.ν. (Γ .p , α. ( p , M ! .p ))where ν is given by the same mediating isomorphism as (25). Our first aim is to show(i) – (iv). First: t.L = t.M T.ν. (Γ .p , α. ( p , M ! .p ))= T.t.ν. (Γ .p , α. ( p , M ! .p ))= T. ( t. Γ .p , t.α. ( p , M ! .p ))= T. ( r.t.p , p )= T. ( r.p.p , p )= T. ( r.p, .p = p and M p.L = M p.M T.ν. (Γ .p , α. ( p , M ! .p ))= M ( p.T ) .ν. (Γ .p , α. ( p , M ! .p ))= M ( s.p ) .ν. (Γ .p , α. ( p , M ! .p ))= M s.M p .ν. (Γ .p , α. ( p , M ! .p ))= M s.p . (Γ .p , α. ( p , M ! .p ))= M s. Γ .p = p , and hence ( M p, t ) .L = 1. Furthermore, L. ( r.p,
1) =
M T.ν. (Γ .p , α. ( p , M ! .p )) . ( r.p, M T.ν. (Γ .r.p, α. (1 , M ! .r.p ))= M T.ν. ( r.r.p, α. (1 , r. ! .p ))= M T.ν. ( r.r.p, α. (1 , r. !))= M T.ν. ( r.r.p, r )= M T.r. ( r.p, r.T. ( r.p, r, so also the second condition for a lift is satisfied.The following calculation shows the third condition: L. ( µ. ( p .p , p .p ) , p ) = M T.ν. (Γ .p , α. ( p , M ! .p )) . ( µ. ( p .p , p .p ) , p )= M T.ν. (Γ .µ. ( p .p , p .p ) , α. ( p , M ! .µ. ( p .p , p .p )))= M T.ν. ( µ. ( M µ.ν. (Γ .p .p , α. ( p .p , M ! .p .p )) , Γ .p .p ) ,µ. ( α. ( p , M ! .p .p ) , α. ( p , M ! .p .p )))= M T.µ. ( ν. ( M µ.ν. (Γ .p .p , α. ( p .p , M ! .p .p )) , α. ( p , M ! .p .p )) ,ν. (Γ .p .p , α. ( p , M ! .p .p )))= µ. ( M T.ν. ( M µ.ν. (Γ .p .p , α. ( p .p , M ! .p .p )) , α. ( p , M ! .p .p )) ,M T.ν. (Γ .p .p , α. ( p , M ! .p .p )))= µ. ( M T.ν. (Γ .p .p , M T.ν. ( α. ( p .p , M ! .p .p ) , α. ( p , M ! .p .p ))) ,L. ( p .p , p ))= µ. ( M T.ν. (Γ .p .p , α. ( T. ( p .p , p ) , M ! .p .p )) , L. ( p .p , p ))= µ. ( M T.ν. (Γ .p , α. ( p , M ! .p )) . ( p .p , T. ( p .p , p )) ,L. ( p .p , p ))= µ. ( L. ( p .p , T. ( p .p , p )) , L. ( p .p , p ))= µ. ( L. ( p .p , s.L. ( p .p , p )) , L. ( p .p , p ))where at = we have used the distributive law, at = we have rewritten the equationof the form (26), at = we have used naturality of µ . At = , we have used thedefinition of L and further that M T.ν. ( M µ.ν. ( M p .M p , M p .M p ) , M p ) = M T.ν. ( M p .M p , M T.ν. ( M p .M p , M p ))which is the image under M of the requirement on T with respect to µ . The step = uses naturality of α (for the square with T ). Then it is a matter of rewriting, and inthe last step we use the equation established at the end of this proof. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 55
For the fourth condition, we again calculate:
M L.ν. (Γ .p , α. ( p , M ! .p )) = M M T.ν.M (Γ .p , α. ( p , M ! .p )) .ν. (Γ .p , α. ( p , M ! .p ))= M M T.ν. ( M Γ .M p , α. ( M p , M M ! .M p )) .ν. (Γ .p , α. ( p , M ! .p ))= M M T.ν. ( M Γ . Γ .p , α. ( α. ( p , M ! .p ) , M M ! . Γ .p ))= M M T.ν. (Γ . Γ .p , Γ .α. ( p , M ! .p ))= M M T. Γ .ν. (Γ .p , α. ( p , M ! .p ))= Γ .M T.ν. (Γ .p , α. ( p , M ! .p ))= Γ .L where = uses the axioms of the strength α with respect to Γ, and the rest arenaturality conditions. This shows that L yields the structure of a naive fibration.Finally, we have s.L = s.M T.ν. (Γ .p , α. ( p , M ! .p ))= T.s.ν. (Γ .p , α. ( p , M ! .p ))= T. ( s. Γ .p , s.α. ( p , M ! .p ))= T. ( p , p )= T, showing that the operations are mutually inverse. (cid:3) To conclude this section, it will be helpful to spell out the notion of morphismbetween naive fibrations explicitly, where the definition is fixed by the notion ofmorphism between the underlying algebras.
Corollary 4.25.
Suppose ( p : E → B, L ) , ( p : E (cid:48) → B (cid:48) , L (cid:48) ) are naive fibrations. Thena morphism of naive fibrations ( p, L ) → ( p (cid:48) , L (cid:48) ) is a commutative square (31) E (cid:48) EB (cid:48) B gfp (cid:48) p such that (32) L. ( g.p , M f.p ) = M g.L (cid:48)
This defines a category
NFib of naive fibrations. Naive fibrations also inherit verticalcomposition of algebras (and can hence be composed). We thus also have a doublecategory N Fib . The Frobenius construction
The goal of this section is to prove a
Frobenius property (see Box 1.3 below) for theAWFS constructed in the previous section in the presence of a certain ‘dual’ AWFSon the same category. This property was also studied for the path object categoriesof Van den Berg and Garner [BG12] (see Definition 3.3.3. loc.cit.). For the ‘dual’AWFS, this section needs some additional assumptions. The additional assumptionsentail that E has a two-sided Moore structure (see Definition A.4 in the Appendix).To the previous section, this adds:(1) a coconnection satisfying the axioms dual to to Γ (with s and t reversed);(2) which satisfies the sandwich axiom : M µ. (Γ ∗ , Γ) = α. (1 , P !): M ⇒ M M ;(3) and the assumption that composition is left and right cancellative: that is,that
M X t × s M X (1 , ∆) (cid:47) (cid:47) M X t × s ( M X ( t,s ) × ( t,s ) M X ) µ. ( p ,p .p ) (cid:47) (cid:47) µ. ( p ,p .p ) (cid:47) (cid:47) M X is an equalizer (and similarly for right cancellative). Since M preserves pull-backs, this remains an equalizer after applying M (so we can apply left can-cellation pointwise).In Section 5.2 we will discuss symmetric Moore structures, for which there is anatural choice of the coconnection Γ ∗ . Note that under the additional structure,there are two AWFSs on E . This yields a definition of naive left fibration . Definition 5.1.
A map p : Y → X together with an arrow L ∗ : M X × X Y → M Y, where
M X × X Y refers to the pullback of p and s (instead of t ), is said to be a naiveleft fibration when it satisfies the conditions:(i) ( M p, s ) .L ∗ = 1;(ii) L ∗ . ( r.p,
1) = r ;(iii) L ∗ . ( µ. ( p .p , p .p ) , p ) = µ. ( L ∗ . ( p .p , t.L ∗ . ( p .p , p )) , L ∗ . ( p .p , p ));(iv) Γ ∗ .L ∗ = M L ∗ . (Γ ∗ .p , α. ( p , M ! .p )). Remark 5.2.
The terminology of left fibrations is adopted from the correspondingnotions (due to Joyal) of left (and right) fibrations in the category of simplicial sets,see e.g. [Lur09], chapter 2. Our notion of effective left and right fibration (developed inSection 6) for categories with Moore structure coincides with left and right fibrationsin simplicial sets in this sense.It follows from the previous sections that naive left fibrations are R -algebras forthe AWFS induced by the coconnection. In particular, Corollary 5.3 gives: FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 57
Corollary 5.3.
Naive left fibrations inherit the structure of a category, denoted
NLFib , whose morphisms are given by commutative squares as in (31) . Similarly, they yielda notion of fibred structure nLFib : E → cartop → S ets isomorphic to R -algebras and a double category N LFib . The equations (ii) – (iii) establish a relation between the two types of AWFS on E . The following lemma exploits this in a way we will need for our formulation of theFrobenius construction in Proposition 5.5 below. In the presence of a two-sided Moorestructure, we will now sometimes refer to naive fibrations as naive right fibrations,to emphasize which one we are talking about. Lemma 5.4.
Suppose p : Y → X has the the structure of a naive right fibration L : M X × X Y → M Y , i.e. they satisfy the conditions of Definition 4.22. Then wealso have: Γ ∗ .L = M L. (Γ ∗ .p , L ) . for the coconnection Γ ∗ .Dually, a naive left fibration ( p : Y → X, L ∗ ) , satisfies Γ .L ∗ = M L ∗ . (Γ .p , L ∗ ) . Proof.
Note that by postcomposing (iv) in Definition 4.22 with
M s we obtain: L = M s.M L. (Γ .p , α. ( p , M ! .p )) . By (pointwise) left cancellation and equation (iv) in Definition 4.22, it suffices toprove:
M µ. (Γ ∗ .L, Γ .L ) = M µ. ( M L. (Γ ∗ .p , L ) , M L. (Γ .p , α. ( p , P ! .p )))But we have M µ. (Γ ∗ .L, Γ .L ) = M µ. (Γ ∗ , Γ) .L = α. (1 , M !) .L = α. ( L, M ! .p ) , as well as M µ. ( M L. (Γ ∗ .p , L ) , M L. (Γ .p , α. ( p , M ! .p ))) = M µ. ( M L. (Γ ∗ .p , M s.M L. (Γ .p , α. ( p , M ! .p ))) , M L. (Γ .p , α. ( p , M ! .p ))) = M L. ( M µ. (Γ ∗ .p , Γ .p ) , α. ( p , M ! .p )) = M L. ( α. ( p , M ! .p ) , α. ( p , M ! .p )) = M L.α. (1 , M ! .p ) = α. ( L, M ! .p ) . In recent literature, the Frobenius property is often formulated as ‘the pullbackof an L -map along an R -map is an L -map’ [BG11][BG11][GS17]. Here L , R are the left and right classes of maps for an algebraic or non-algebraic weak-factorization system: D AE B i ∈ L∈ L p ∈ R The property is of interest because it is closely related to right-properness ofthe model structure, if the factorization system is part of one (see [GS17])and dependent products for a model of type theory (as in Section 7 below).But the Frobenius property of this section is more abstract. The class of rightmaps R in the above diagram is replaced by the right maps from a different,but closely related, AWFS. The actual Frobenius property is extracted as aconsequence for symmetric Moore structures in Corollary 5.11.
Box 1.3.
The Frobenius propertyThe second statement is dual to the first. (cid:3)
The Frobenius construction.
The following proposition contains our defini-tion of the
Frobenius construction in the current context. As a property of an AWFSresulting from a Moore structure, the proposition is comparable to the
Frobeniusproperty of Garner-van den Berg [BG12]. See also Box 1.3.
Proposition 5.5. (Frobenius construction)
Suppose ( i, j, H ) is an HDR and A (cid:48) i (cid:48) (cid:15) (cid:15) p (cid:48) (cid:47) (cid:47) A i (cid:15) (cid:15) E p (cid:47) (cid:47) B is a pullback square in which p : E → B is a naive left fibration. Then i (cid:48) can beextended to an HDR such that the square becomes a morphism of HDRs. Remark 5.6.
The diagram above will not be a cartesian morphism of HDRs, ingeneral.
Proof.
Let L ∗ be the naive left fibration structure on p . We have a map j : B → A and a homotopy H : B → M B with j.i = 1 , s.H = 1 , t.H = i.j and Γ .H = M H.H . In
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 59 addition, we have a pullback diagram of the form A (cid:48) i (cid:48) (cid:15) (cid:15) p (cid:48) (cid:47) (cid:47) A i (cid:15) (cid:15) E p (cid:47) (cid:47) B. Write H (cid:48) = L ∗ . ( H.p, E → M E . Then s.H (cid:48) = s.L ∗ . ( H.p,
1) = p . ( H.p,
1) = 1 and p.t.H (cid:48) = p.t.L ∗ . ( H.p,
1) = t.M p.L ∗ . ( H.p,
1) = t.p . ( H.p,
1) = t.H.p = i.j.p, and therefore there is a map j (cid:48) : E → A (cid:48) with p (cid:48) .j (cid:48) = j.p and i (cid:48) .j (cid:48) = t.H (cid:48) . We willfirst show that j (cid:48) .i (cid:48) = 1 and M H (cid:48) .H (cid:48) = Γ .H (cid:48) .To see j (cid:48) .i (cid:48) = 1, we calculate i (cid:48) .j (cid:48) .i (cid:48) = t.H (cid:48) .i (cid:48) = t.L ∗ . ( H.p, .i (cid:48) = t.L ∗ . ( H.p.i (cid:48) , i (cid:48) )= t.L ∗ . ( H.i.p (cid:48) , i (cid:48) )= t.L ∗ . ( r.i.p (cid:48) , i (cid:48) )= t.L ∗ . ( r.p.i (cid:48) , i (cid:48) )= t.L ∗ . ( r.p, .i (cid:48) = t.r.i (cid:48) = i (cid:48) = i (cid:48) . . To prove
M H (cid:48) .H (cid:48) = Γ .H (cid:48) , we compute: M H (cid:48) .H (cid:48) = M ( L ∗ . ( H.p, .L ∗ . ( H.p,
M L ∗ .M ( H.p, .L ∗ . ( H.p,
M L ∗ . ( M H.M p, .L ∗ . ( H.p,
M L ∗ . ( M H.M p.L ∗ , L ∗ ) . ( H.p,
M L ∗ . ( M H.p , L ∗ ) . ( H.p,
M L ∗ . ( M H.H.p, L ∗ ( H.p,
M L ∗ . (Γ .H.p, L ∗ . ( H.p,
M L ∗ . (Γ .p , L ∗ ) . ( H.p, .L ∗ . ( H.p, .H (cid:48) . Here we have used the identity of Lemma 5.4.
It remains to check that the square is a moprhism of HDRs. However, we have p (cid:48) .j (cid:48) = j.p , by construction, and M p.H (cid:48) = M p.L ∗ . ( H.p, p . ( H.p,
H.p. (cid:3)
Lemma 5.7.
The Frobenius construction of Proposition 5.5 is stable under compo-sition of naive left fibrations p as well as composition of HDRs i .Proof. Consider a picture as follows: A i (cid:15) (cid:15) q (cid:47) (cid:47) A i (cid:15) (cid:15) q (cid:47) (cid:47) A i (cid:15) (cid:15) E p (cid:47) (cid:47) E p (cid:47) (cid:47) E . Then we have H = L ∗ p . ( H .p , H = L ∗ p . ( H .p , H ∗ = L ∗ p .p . ( H .p .p , H ∗ = H , which we can do as follows: H ∗ = L ∗ p .p . ( H .p .p , L ∗ p . ( L ∗ p . ( p , H .p .p ) , L ∗ p . ( L ∗ p . (1 , H .p ) .p ) , L ∗ p . ( H .p , H . Now consider a picture as follows: E p (cid:47) (cid:47) i (cid:48) (cid:15) (cid:15) A i (cid:15) (cid:15) E p (cid:47) (cid:47) i (cid:48) (cid:15) (cid:15) A i (cid:15) (cid:15) E p (cid:47) (cid:47) A . Then we have: H (cid:48) = L ∗ p . ( H .p , H (cid:48) = L ∗ p . ( H .p , H (cid:48) = H (cid:48) ∗ H (cid:48) H ∗ = L ∗ p . (( H ∗ H ) .p , H (cid:48) and H ∗ . So here we go: H ∗ = L ∗ p . (( H ∗ H ) .p , L ∗ p . ( µ. ( H .p , M i .H .j .p ) , µ. ( L ∗ p . ( H .p , , L ∗ p . ( M i .H .j .p , t.L ∗ p ( H .p , µ. ( H (cid:48) , L ∗ p . ( M i .H .j .p , t.H (cid:48) , ))= µ. ( H (cid:48) , L ∗ p . ( M i .H .p .j (cid:48) , i (cid:48) .j (cid:48) ))= µ. ( H (cid:48) , M i (cid:48) .L ∗ p ( H .p , .j (cid:48) )= µ. ( H (cid:48) , M i (cid:48) .H (cid:48) .j (cid:48) )= H (cid:48) ∗ H (cid:48) = H (cid:48) and the proof is finished. (cid:3) For an HDR ( i : A → B, j, H ), we call a morphism of HDRs as defined by (theproof of) Proposition 5.5 a
Frobenius morphism of HDRs.
Lemma 5.8. (Pullback stability of Frobenius construction)
The Frobenius construc-tion of Proposition 5.5 defines a functor : ( − ) ∗ ( − ): NLFib × E HDR → HDR where the domain is the pullback of the domain functors to E .As a consequence, the pullback (Corollary 4.11) in HDR of a Frobenius morphismalong a morphism of HDRs is again a Frobenius morphism of HDRs.Proof.
Suppose ( a, b ): i → i is a morphism of HDRs, ( q , p ): i (cid:48) → i is a Frobeniusmorphism of HDRs, and ( f, b ): p → p is a morphism of naive left fibrations, as in the solid part of the following diagram:(33) E A E A F B F B ei (cid:48) ai i (cid:48) q i f p bp It is enough to prove that the Frobenius construction applied to the back squareinduces a unique morphism of HDRs i (cid:48) → i (cid:48) on the left side of the cube.So all that needs to be verified is that ( e, f ): i (cid:48) → i (cid:48) induced by the pullbackis a morphism of HDRs. Denoting their respective HDR structure by H (cid:48) , H (cid:48) , anddenoting the naive left fibrations by ( p , L p ) and ( p , L p ), we compute: H (cid:48) .f = L ∗ p . ( H .p , .f = L ∗ p . ( H .p .f, f )= L ∗ p . ( H .b.p , f ) = L ∗ p . ( M b.H .p , f )= M f.L ∗ p . ( H .p ,
1) =
M f.H (cid:48) Where we have used that the bottom face is a morphism of naive left fibrations. ByCorollary 4.9, it follows that the left face is a morphism of HDRs.For the last statement, it is easy to see that the cube is a pullback square ofmorphisms of HDRs. (cid:3)
Symmetric Moore structure.
In the appendix, we have defined a particularclass of Moore structures for which there is a natural choice of the coconnectionΓ ∗ . We call such Moore structures symmetric (Definition A.5). A symmetric Moorestructure comes equipped with a natural transformation τ : M ⇒ M such that:(i) For every X , τ X : M X → M X is an internal, idempotent identity-on-objectsfunctor between the category
M X (given by the Moore structure) and the op-posite category on
M X . In particular s.τ = t and t.τ = s .(ii) The natural transformation Γ ∗ : = τ M .M ( τ ) . Γ .τ satisfies the conditions of a co-connection from the beginning of this section. Remark 5.9.
The first condition implies that Γ ∗ defined in the above way satisfiesthe dual axioms of a connection. Proposition 5.10.
In a category with symmetric Moore structure, HDRs coincidefor both Moore structures. That is, there is a double functor over Sq ( E ) between thetwo categories of HDRs which is an equivalence. Hence the following can be concluded: FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 63 (i) There is an isomorphism between the notions of fibred structure: • Naive (right) fibrations nFib : E cart → → S ets, • Naive left fibrations nLFib : E cart → → S ets ; (ii) There is an equivalence of categories NFib ∼ = NLFib which is prescribed on objects by (i);(iii) There is an equivalence of double categories N Fib ∼ = N LFib . whose vertical restriction is prescribed by (ii).Proof. If ( i : A → B, j, H ∗ ) is a ‘left-HDR’, so for the dual structure given by Γ ∗ , thenlet H : = τ B .H ∗ . We claim that ( i, j, H ) is an HDR. The first requirements are easyto check. For the condition on Γ, we have:Γ .H = Γ .τ.H ∗ = M ( τ ) .τ. Γ ∗ .H ∗ = M ( τ ) .τ.M ( H ∗ ) .H ∗ = M ( τ ) .M ( H ∗ ) .τ.H ∗ = M ( τ.H ∗ ) .τ.H ∗ = M ( H ) .H Functoriality with respect to squares is easy to see, and for vertical composition asgiven by (27) we have:( τ.H ) ∗ ( τ.H ) = µ. ( τ.H , M i .τ.H .j )= µ. ( τ.H , τ.M i .H .j )= τ.µ. ( H , M i .H .j )= τ. ( H ∗ H )Note that this proposition uses the first condition on τ entirely. Of course this argu-ment dualizes, and clearly the two operations are inverse. The induced equivalencesfollow from the induced equivalences in the AWFS. (cid:3) Corollary 5.11. (Frobenius for symmetric Moore structures)
In categories with sym-metric Moore structure, there is a pullback functor ( − ) ∗ ( − ): NFib × E HDR → HDR given by factoring the Frobenius construction of Lemma 5.8 through the equivalencebetween naive left and right of Proposition 5.10. Mould squares and effective Kan fibrations
Mould squares.
In this section, we connect the two algebraic weak factorisationsystems coming from a dominance and a Moore structure, under some assumptionscombining the assumptions of Sections 3 and 4. First, we assume that E is nowa finitely cocomplete, locally cartesian closed category with Moore structure (recallthe remarks in the introduction of Section 2). In addition, we assume that E comesequipped with a dominance as in Section 3, such that: • Σ is closed under binary unions of subobjects; • Σ contains every initial arrow 0 → A in E .The coalgebras coming from the dominance are called cofibrations . Lastly, we alsomake the combining assumption that every trivial Moore path r X : X → M X is a cofibration . Below, we will combine cofibrations with HDRs to present a newnotion of fibred structure, called an effective fibration . Definition 6.1. A mould square is a cartesian morphism of HDRs in the fibre abovea cofibration, as in the following diagram:(34) A (cid:48) B (cid:48) A B m m (cid:48) ii (cid:48) where i , i (cid:48) are HDRs and m is a cofibration (so m (cid:48) is also a cofibration). Lemma 6.2.
The pullback of a mould square along an arbitrary morphism of HDRsis again a mould square.Proof.
This follows directly from Corollary 4.18 and the fact cofibrations are stableunder pullback. (cid:3)
Mould squares are the squares of a double category whose objects are objects of E ,horizontal morphisms are HDRs, and vertical morphisms are cofibrations. However,it turns out that we need a little bit more. First, we motivate this.6.1.1. A new notion of fibred structure.
Given a double category L : L → Sq ( E ) over Sq ( E ), there is another way to define a notion of lifting structure for a map p : Y → X . When r is a cartesian natural transformation, this is equivalent to r : 1 → M r is not cartesian in some other examples. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 65
Triple categories are not much harder to understand than double categories.We can define a small triple category as an internal category in the categoryof small double categories. This definition can be unfolded (and hence ex-tended to include large triple categories) in the same way as in Section 2.2.Briefly, they extend double categories in that there is an additional type of1-dimensional morphisms, perpendicular morphisms, and hence three differenttypes of squares, between each pair of distinct 1-dimensional morphism types.In addition, cubes are morphisms between xy -squares which compose in theperpendicular direction, and come with additional ‘pointwise’ compositions forthe horizontal and vertical direction. So a cube looks like: A (cid:48) B (cid:48) A BC (cid:48) CD (cid:48) D ( xy ) ⇒ ( y z ) ⇒ ( x z ) ⇒ Note we are using the convention here that the three axes in 3-dimensionalspace are named as follows: xyz
A cube admits composition from three different sides. The axioms guaranteethat composition of a given combination of squares and cubes is independent ofthe order in which compositions are taken along the different dimensions. Thestandard example of a triple category is the category
Cube ( E ) for a category E , where objects are objects in E , all morphisms are given by morphisms in E ,all squares are commutative squares, and cubes are commutative cubes. Box 1.4.
Triple categoriesNamely, given a square σ = ( f, g ): v (cid:48) → v in L and a commutative diagram in E as in the solid part of the diagram below: A (cid:48) A YB (cid:48)
B X I ( v (cid:48) ) I ( f ) I ( g ) I ( v ) ab ph φ a,b ( f, g, h ) ,where the square on the left is the image of σ under I , there should be a filler φ u,v ( f, g, h ) as drawn such that the diagram commutes. In other words, we requirethat the ‘partial filler’ h can be extended.In this section, we define uniform fibrations as maps equipped with this type oflifting structure with respect to the mould squares of Definition 6.1. Yet we also needto impose compatibility conditions with respect to certain morphisms between mouldsquares. This can be done by defining this new notion of lift as a lifting structurewith respect to a triple category . We have put a brief introduction to triple categoriesin Box 1.4. The following definition spells out what it means to have a right liftingstructure with respect to a triple category over Cube ( E ). Definition 6.3.
Suppose L is a triple category and I : L → Cube ( E ) is a triplefunctor, and suppose p : Y → X is a morphism in E . Then a right lifting structure for p with respect to I is a family of fillers for each diagram: A (cid:48) A YB (cid:48)
B X I ( v (cid:48) ) I ( f ) I ( g ) I ( v ) ab ph φ a,b ( f, g, h ) ,such that the following conditions hold: Horizontal:
When ( f (cid:48) , g (cid:48) ), ( f, g ) is a horizontally composable pair of xy -squares,and we are given a commutative diagram: A (cid:48)(cid:48) A (cid:48) A YB (cid:48)(cid:48) B (cid:48) B X I ( f (cid:48) ) I ( g (cid:48) ) I ( v (cid:48)(cid:48) ) I ( v (cid:48) ) I ( f ) I ( g ) I ( v ) ab ph then we have: φ a,b ( f.f (cid:48) , g.g (cid:48) , h ) = φ a,b ( f, g, φ a.I ( f ) ,b.I ( g ) ( f (cid:48) , g (cid:48) , h )) FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 67
Vertical:
If we have a vertically composable pair of xy -squares and a diagram: A (cid:48) A YB (cid:48) BC (cid:48) C X I ( w (cid:48) ) I ( w ) I ( v (cid:48) ) I ( k ) I ( f ) I ( g ) I ( v ) ac ph ,then we have: φ a,c ( f, k, h ) = φ φ a,c.I ( v ) ( f,g,h.I ( v (cid:48) )) ,c ( g, k, h ) Perpendicular:
For cubes , the condition asks that for the image of a cubebetween xy -squares: A (cid:48) AB (cid:48) BC (cid:48) C XD (cid:48)
D X pcdhI ( f ) I ( g ) I ( k ) I ( l ) I ( v (cid:48) ) I ( v ) I ( w (cid:48) ) I ( w ) u (cid:48) r u we have u.φ c,d ( k, l, h ) = φ d.u,c.u (cid:48) ( f, g, h.r ) . Remark 6.4.
As for double categories, there is a symmetry in the definition of atriple category, namely the choice of ‘top level’ domain and codomain between cubes(the same goes for squares), which could be any of the three xy , yz , xz directions.In Definition 6.3 it is assumed that cubes are morphisms between xy -squares, justlike the definition in Box 1.4. The definition of lifting structure takes this choice as astarting point. We do note that the definition of right lifting structure is is symmetricin x and y , i.e. we could swap the horizontal and vertical morphisms. Remark 6.5.
The fillers in Definition 6.3 are just ordinary fillers with respect toarrows B (cid:48) + A (cid:48) A → B induced by the pushout of the square. This is similar to theway generating trivial cofibrations are defined by Gambino and Sattler [GS17]. Yet,formulating the horizontal, vertical and perpendicular conditions in terms of pushoutsis very cumbersome. It was not until we arrived at mould squares that we understoodwhat uniformity conditions for Kan fibrations should look like. The present form isused very intensely in part 2 of this paper on simplicial sets. Proposition 6.6.
Right lifting structures with respect to a triple category are closedunder pullback. That is, there is a fibred structure (Definition 2.1): fib : E → cart → S ets which associates to an arrow in E the set of right lifting structures on that arrow.Further, right lifting structures with respect to a triple category also form a doublecategory, analogous to Definition 2.9.Proof. This is straightforward and left to the reader. (cid:3)
Effective fibrations.
We can now give the definition central to this paper,which combines all previous sections.
Definition 6.7. An effective fibration in a Moore category E equipped with a dom-inance is a morphism p : Y → X equipped with a right lifting structure with respectto the following triple category:(i) Objects are the objects of E .(ii) Horizontal morphisms are HDRs.(iii) Vertical morphisms are cofibrations.(iv) Perpendicular morphisms are morphisms in E .(v) xy -squares are mould squares.(vi) xz -squares are morphisms of HDRs.(vii) yz -squares are morphisms of cofibrations, i.e. pullback squares.(viii) Cubes are pullback ‘squares’ of a mould square along a morphism of HDRs(which always yields a mould square as per Lemma 6.2).Note that cubes in this triple category are unique for a given boundary of a cube,which consists of six faces.We denote the fibred structure (Proposition 6.6) of effective Kan fibrations by effFib , the double category of effective Kan fibrations by E ffFib (over Sq ( E )), andits category of vertical morphisms and squares EffFib (over E ).The following lemma is a motivator and sanity check for the terminology we haveused for effective fibrations. Lemma 6.8. (i) There is a natural transformation between notions of fibred struc-tures effFib → nFib ; (ii) There is a functor EffFib → NFib which is prescribed on objects by (i);(iii) There is a double functor E ffFib → N Fib whose vertical restriction is prescribed by the functor (ii).
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 69
Proof.
We use the assumption from the beginning of this section, that every objectis cofibrant, i.e. every 0 → A is contained in the dominance. Suppose p : Y → X is aneffective fibration. We will show that p can be equipped with a right lifting structurewith respect to the double category of HDRs. Given an HDR i : A → B , we can definethe structure as the family of fillers:0 0 YA B X i v pu φ u,v ( i ) ,By our assumption, the square on the left is indeed a mould square. It is easy to seethat the horizontal condition on effective Kan fibrations implies the vertical conditionof a right lifting structure of Section 2.2. Similarly, the perpendicular conditionimplies the vertical condition. It follows that the family φ gives p the structure of an R -algebra. It is left to the reader to verify the three types of functoriality. (cid:3) Trivial fibrations.
Recall from Section 3 that we named the right class withrespect to the double category Σ of cofibrations (or the dominance) trivial fibrations (Definition 3.6). Similar to effective fibration, there is a notion of effective trivialfibration , induced by the following triple category:(i) Objects are the objects of E ;(ii) Horizontal and vertical morphisms are cofibrations;(iii) Perpendicular morphisms are morphisms in E ;(iv) xy -squares are pullback squares of cofibrations;(v) xz -squares are commutative squares of cofibrations and morphisms in E ;(vi) yz -squares are pullback squares of a cofibration along a morphism in E ;(vii) Cubes are formed by pulling back an yz square along an xz square.First, we observe the triple category of mould squares is a sub triple category ofthis triple category, since: Lemma 6.9.
Every HDR is a cofibration.Proof.
This is straightforward using Lemma 4.7, under the ruling assumption thatevery r X : X → M X is a cofibration. (cid:3)
In the next proposition, we have adopted the familiar notation for effective trivialfibrations.
Proposition 6.10. (i) There is an isomorphism between notions of fibred struc-tures trivFib ∼ −→ effTrivFib ; (ii) There is an equivalence TrivFib ∼ −→ EffTrivFib which is prescribed on objects by (i);(iii) There is a double equivalence T rivFib ∼ −→ E ffTrivFib whose vertical restriction is prescribed by the functor (ii).Proof. The non-trivial part is to show that every trivial fibration can be equippedwith the structure of an effective trivial fibration in some functorial (isomorphic) way.So suppose p : Y → X is a trivial fibration. Every lifting problem with respect to amould square factors through a coproduct as follows: A (cid:48) B (cid:48) YA + A (cid:48) BA B X m i h + uv p Observe that A + A (cid:48) B (cid:48) (cid:44) → B is a cofibration under the prevailing assumption thatthese are closed under binary unions, mould squares are always pullback squaresand the fact that E is coherent (here we assume locally cartesian closed and finitelycocomplete). Hence there exists a filler B → Y as drawn. We need to check thatit satisfies the horizontal, vertical and perpendicular conditions. For the horizontalcondition, the lifting problem factors as follows: A (cid:48) B (cid:48) C (cid:48) YA + A (cid:48) B (cid:48) A + B (cid:48) C (cid:48) A B B + B (cid:48) C (cid:48) C X ( ∗ ) where we observe that the square ( ∗ ) is both a pullback and a pushout. Hence thehorizontal condition for trivial fibration applies, and the lift B → Y is determinedby the lift B + B (cid:48) C (cid:48) → Y . By the vertical condition, the latter is in turn compatiblewith the subsequent lift C → Y and equal to the lift with respect to A + B (cid:48) C (cid:48) (cid:44) → C .It follows that the dashed fillers make the whole diagram commute, which proves thehorizontal condition. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 71
The vertical condition can be proven similarly: A (cid:48)(cid:48) B (cid:48)(cid:48) YA (cid:48) + A (cid:48)(cid:48) B (cid:48)(cid:48) A (cid:48) B (cid:48) A + A (cid:48)(cid:48) B (cid:48)(cid:48) A + A (cid:48) B (cid:48) A B X ( ∗ )Again, the square ( ∗ ) is both pullback and pushout. The argument is now the sameas in the horizontal case.The perpendicular case is easy and follows from the horizontal condition for trivialfibrations.It is left to the reader to convince themselves that this indeed induces the statedisomorphism and equivalences. (cid:3) From the above, we obtain:
Corollary 6.11. (i) There is a natural transformation between notions of fibredstructures trivFib → effFib ; (ii) There is a functor TrivFib → EffFib which is prescribed on objects by (i);(iii) There is a double functor T rivFib → E ffFib whose vertical restriction is prescribed by the functor (ii).Proof. This follows directly from the existence of a triple functor from the triplefunctor of mould squares to the triple category in this subsection. (cid:3)
Right, left and Kan fibrations.
Having established the definition of effectivefibration, we turn to the situations of Section 5. There, we studied the AWFS for two-sided and symmetric Moore structures. Naturally, the two-sided setting admits a dualnotion of fibred structure which we call effective left fibration (for the terminology,recall Remark 5.2). In the symmetric case, we obtain:
Corollary 6.12.
Suppose the Moore structure on E is symmetric as in as in Sec-tion 5.2. Then:(i) There is an isomorphism between the notions of fibred structure: • Effective (right) fibrations effFib : E cart → → S ets, • Effective left fibrations effLFib : E cart → → S ets ; (ii) There is an equivalence of categories EffFib ∼ = EffLFib which is prescribed on objects by (i);(iii) There is an equivalence of double categories E ffFib ∼ = E ffLFib whose vertical restriction is prescribed by (ii).Proof. This essentially follows from Lemma 6.8 and Proposition 5.10. (cid:3)
Definition 6.13.
In a category E with a symmetric Moore structure satisfying theconditions of the beginning of this section, the notion of fibred structure, the category,or double category of the items of Corollary 6.12 are called effective Kan fibrations . FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 73
7. Π -types
This section contains the main result of this part of the paper. Described inmore familiar terms, the result gives a constructive proof of the fact that when X iseffectively fibrant, and A is any other object, then the exponential X A is effectivelyfibrant (see Remark 7.2). The statement of the proposition is a more general, fibredversion of this fact, which is a classic result on Kan fibrations in simplicial sets (seee.g. [May67]).We assume that E satisfies the conditions from the beginning of Section 6 and thatthe Moore structure on E is two-sided. In particular, we do not (yet) assume thatthe Moore structure on E is symmetric. Theorem 7.1. If f : Y → X is a naive left fibration and g : Z → Y is an effectivefibration, then the dependent product Π f ( g ) is also an effective Kan fibration. Moreprecisely, the pullback along a naive left fibration f ∗ : EffFib X → EffFib Y whichtakes effective fibrations with codomain X to effective Kan fibrations with codomain Y has a right adjoint f ∗ : EffFib Y → EffFib X defined by a functor Π ( − ) ( − ): NLFib × E EffFib → EffFib where the domain is the pullback of the domain and codomain functors.Proof.
Assume f is a naive left fibration and g : Z → Y is an effective fibration. Wehave to show that Π f ( g ) is an effective fibration. So imagine we have a situation likethis: A (cid:48) B (cid:48) WA B XZY fgi (cid:48) i Π f ( g ) in which the left hand square is a mould square. The construction starts by takingthe pullback of f along A → X , which yields a naive left fibration, and subsequently constructing a ‘Frobenius’ cube like in (33): A (cid:48) B (cid:48) WA B XC (cid:48) D (cid:48) ZC D Y fg ∃ i (cid:48) i Π f ( g ) So the front and back squares are mould squares and the bottom and top are Frobeniusmorphisms of HDRs. The maps D (cid:48) → Z and C → Z are induced by the adjunction f ∗ (cid:97) Π f . Since g is an effective fibration, there is a map D → Z making everythingcommute, which we can transpose back to a map B → W giving a filler.It remains to check the three compatibility conditions, horizontal, vertical and per-pendicular, for this filler. The horizontal and vertical conditions follow fairly directlyfrom the naturality condition for adjunctions together with Lemma 5.7. Indeed, forthe horizontal condition: A (cid:48) B (cid:48) C (cid:48) WA B C XD (cid:48) E (cid:48) F (cid:48) ZD E F Y fg ∃ Π f ( g ) To find a map C → W we can either transpose D → Z and then push forward in onego as F → Z and then transpose back. This should coincide with: first obtaining E → Z , transpose, then transpose back and compute F → Z and then transpose. Itis clear that this will be the same as the other procedure. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 75
The vertical condition is similar: A (cid:48)(cid:48) B (cid:48)(cid:48) WA (cid:48) B (cid:48) A B XC (cid:48)(cid:48) D (cid:48)(cid:48) ZC (cid:48) D (cid:48) C D Y g f Π f g Given a map A → W one can transpose to C → Z , then push forward to D → W and then transpose back. Using the vertical condition on g , this can also be donein two steps. Namely, by first lifting the restricted arrow C (cid:48) → Z to D (cid:48) → Z , thentransposing, and repeating the construction for the bottom cube. This works becausethe square D (cid:48) B (cid:48) BD is a pullback, hence we return to the same diagram (same arrows)in the two-step version.For the perpendicular condition, it helps to reduce dimensions by one by drawingHDRs as a point, as in the following diagram: i i Wi (cid:48) i (cid:48) Zi i Xi (cid:48) i (cid:48) Y f (cid:48)(cid:48) f (cid:48) bb (cid:48) g f Π f ( g ) Here the open arrowtips indicate mould squares, and the double arrowtips indicateFrobenius morphisms of HDRs. In this picture, lifting can be viewed as completinga ‘partial’ arrow i → W to a total arrow. Now suppose the left square on the backis a cube between mould squares.We can complete the diagram as drawn by forming a pullback cube in the category HDR . Since cubes in the triple category are determined by their boundary, the leftsquare on the front is a cube between mould squares. Hence the filler i (cid:48) → Z inducedby g makes the triangle with the filler i (cid:48) → Z commute. The former uniquely determines the filler i → W , and the latter uniquely determines the filler i → W .So these also make the triangle in the back commute and we are done.Note that the crucial part of this argument is Lemma 5.8 since without it, it wouldnot have been possible to work entirely inside the category HDR , and it would nothave been clear that the two ways to construct the diagram (starting with f (cid:48) vs.starting with b ) would yield the same 4-dimensional cube, with the same relevantHDR structures, in the underlying category. (cid:3) Remark 7.2.
To relate the previous theorem to the statement in the beginning ofthis section, consider that in a category with Moore structure, every object is naivelyfibrant. Indeed, any terminal arrow A → A × M → A given by projection. So every A → X A : = Π A → X × A → p : X × A → X is.For a category with symmetric Moore structure, then, the theorem implies thateffective Kan fibrations are closed under pushforward. Corollary 7.3. (Kan Dependent Products)
Suppose E has a symmetric Moore struc-ture, as in Section 5.2. Then the dependent product Π g f of an effective Kan fibration f along an effective Kan fibration g is again an effective Kan fibration. More pre-cisely, there is a functor Π ( − ) ( − ): EffFib × E EffFib → EffFib as in Theorem 7.1 such that Π g ( − ) is right adjoint to pullback along g .Proof. This follows by precomposing the functor of Theorem 7.1 with the equivalencein Corollary 6.12 and the functor in Lemma 6.8. (cid:3)
HAPTER 2
Simplicial sets
778 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS Effective trivial Kan fibrations in simplicial sets
Whereas in the first part we derived the existence of Π-types in an axiomatic settingbased on a suitable combination of a dominance and a symmetric Moore structure,this second part will be entirely devoted to one particular example: simplicial sets.To show that the category of simplicial sets is indeed an example, we will first haveto choose both a dominance and a symmetric Moore structure on simplicial sets, andthe first two sections of this second part will do exactly that. Indeed, in this sectionwe will choose a dominance and in the next section we will choose a symmetric Moorestructure. After that, we will study the resulting HDRs and effective Kan fibrationsin more detail. In particular, we will show that the effective Kan fibrations are alocal notion of fibred structure and that in a classical metatheory the maps whichcan be equipped with the structure of an effective Kan fibration are precisely thosemaps which have the right lifting property against horn inclusions.But before we get into that, let us first choose and study a suitable dominance onsimplicial sets. As we have seen in Section 3, dominances induce AWFSs. We will callthe coalgebras for the comonad of the induced AWFS effective cofibrations , while thealgebras for the monad will be called the effective trivial Kan fibrations . The mainresults of this section will be that being an effective trivial Kan fibration is a localnotion fibred structure, and that (in a classical metatheory) a map can be equippedwith structure of an effective trivial Kan fibration if and only if it has the right liftingproperty against boundary inclusions.8.1.
Effective cofibrations.
Traditionally, the cofibrations in simplicial sets aresimply the monomorphisms. If one works in a constructive metatheory, however, itis important to add a decidability condition.
Definition 8.1.
In the category of simplicial sets we will call m : B → A an (effective)cofibration if each m n : B n → A n is a complemented monomorphism in the subobjectlattice of A n . In other words, for each a ∈ A n we can decide whether there is anelement b ∈ B n such that m n ( b ) = a or not. Lemma 8.2.
The following are equivalent for a sieve S ⊆ ∆ n :(1) The inclusion i : S ⊆ ∆ n is an effective cofibration.(2) The sieve is generated by a finite set of monos ∆ m → ∆ n .(3) The sieve is generated by a finite set of maps.Proof. To be clear, by saying that a sieve S is generated by a set of maps I , we meanthat S is the closure of I under precomposition with arbitrary maps, and a set isfinite if it can be put in bijective correspondence with some initial segment of thenatural numbers. In that case, the implications (2) ⇒ (3) ⇒ (1) are obvious, so itremains to show that (1) ⇒ (2).But since every map in ∆ factors as an epi followed by a mono (in a unique way),and every epi splits, every sieve is generated by its monomorphisms. But since there FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 79 are only finitely many monos with codomain ∆ n , a cofibrant sieve contains onlyfinitely many monos. (cid:3) Definition 8.3.
We will refer to the sieves satisfying any of the equivalent conditionsin the previous lemma as the cofibrant sieves .Because a monomorphism in ∆ with codomain [ n ] is completely determined by itsimage, a cofibrant sieve can be thought of as a subsimplicial complex of ∆ n , that is,a family of inhabited subsets of { , , . . . , n } closed under subsets. Theorem 8.4.
The effective cofibrations in simplicial sets form a dominance.Proof.
The cofibrations are clearly closed under pullback and composition, so we onlyneed to prove that there is a cofibration 1 → Σ such that any other can be obtainedas a pullback of that one in a unique way. We putΣ n : = { S ⊆ ∆ n : S cofibrant sieve } . (Note that Σ n is finite, so that Σ n is a set even in a predicative metatheory like CZF .) Since cofibrant sieves are stable under pullback along α : ∆ m → ∆ n , thisdefines a simplicial set. That is, we define the action on Σ by the following formula: S · α = { β : [ k ] → [ m ] : α.β ∈ S } . In addition, there is a natural transformation (cid:62) : 1 → Σ obtained by picking themaximal sieve at each level. This map classifies the cofibrations in that for anycofibration m : B → A the map µ : A → Σ defined by µ n ( a ) = { α : [ m ] → [ n ] : a · α ∈ B m } turns B A Σ m (cid:62) µ into a pullback. Also, the map µ is easily seen to be unique with this property. (cid:3) Remark 8.5.
Note that the cofibrant subobjects form a sub-Heyting algebra of thefull subobject lattice. In particular, the cofibrant sieves are closed under all thepropositional operations: not only ∧ , (cid:62) , but also ⊥ , ∨ and → . To see that they areclosed under implication, for instance, note that for sieves S, T ⊆ ∆ n , we have α : [ m ] → [ n ] ∈ ( S → T ) ⇐⇒ ( ∀ β : [ k ] → [ m ]) ( α.β ∈ S ⇒ α.β ∈ T ) . Because maps in ∆ factor as an epi followed by a mono, and epis split, we only needto check the condition on the right for monos β . So if S and T are cofibrant, thecondition on the right is decidable and S → T is cofibrant as well. Effective trivial Kan fibrations.
Since the effective cofibrations in simplicialsets form a dominance, they are the left class in an algebraic weak factorisationsystem. The members of the right class will be referred to as the effective trivial Kanfibrations . From the work by Bourke and Garner (recapitulated in Section 2), weknow that these can be defined as the maps which come with a compatible system oflifts against a large double category: one where the vertical maps are the cofibrationsand the squares are pullback squares. Our first goal in this subsection is to show thatwe can restrict attention to a small subdouble category.Indeed, let C be the following small double category: • Objects are cofibrant sieves S ⊆ ∆ n . • Horizontal maps from S ⊆ ∆ n to T ⊆ ∆ m are maps α : ∆ n → ∆ m such that T · α = S . • Vertical maps are inclusions of cofibrant sieves S ⊆ S ⊆ ∆ n . • Squares are pullback diagrams of the form S T S T αα such that both horizontal maps are labelled with the same α .Clearly, there is an inclusion of double categories from C to the large double categoryof cofibrations. Proposition 8.6.
The following notions of fibred structure are isomorphic: • Having the right lifting property against the large double category of cofibra-tions (that is, to be an effective trivial Kan fibration). • Having the right lifting property against the small double category C .More precisely, the morphism of notions of fibred structure induced by the inclusionof C in the large double category of cofibrations is an isomorphism.Proof. Assume p : Y → X has the right lifting property against the small doublecategory C , and imagine that we have a lifting problem of the form: B YA X. m pl
Suppose a ∈ A n is arbitrary and we pull back m along a : ∆ n → A : S B Y ∆ n A X. m pa
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 81
Since the pullback is a cofibrant sieve, we find an element y ∈ Y n filling the outerrectangle, and we put l n ( a ): = y . Note that this definition is forced, because the lefthand square in the diagram above is a square in the large double category. Note alsothat l is going to be a natural transformation because of the horizontal conditioncoming from the pullback squares in C : S · α S B Y ∆ m ∆ n A X. m pα a
Next, let us check that in case m is a vertical map coming from the small doublecategory, the new lift l agrees with the one coming from the fact that p has the rightlifting property against C : for note that if S ⊆ T ⊆ ∆ n are cofibrant sieves and α : ∆ m → ∆ n ∈ T , then T · α = ∆ m and the left hand square in S · α S Y ∆ m T X py is a square in the double category C . So both the lift T → Y we have constructedand the one coming from the fact that p has the right lifting property against C send α to the lift y .It is now easy to see that the constructed lifts satisfy both the horizontal andvertical conditions with respect to the large double category, thus finishing the proof. (cid:3) We will now cut down things even further. In fact, the lifting structure against C is completely determined by its lifts against the boundary inclusions, as we will nowshow. Lemma 8.7.
Suppose p : Y → X has two lifting structures against the small doublecategory C . If these two lifting structures agree on the lifts against the boundaryinclusions, then they agree on all vertical maps.Proof. Let S ⊆ T ⊆ ∆ n be cofibrant sieves. Then this inclusion can be decomposedas S = S ⊆ S ⊆ S ⊆ . . . ⊆ S k = T ⊆ ∆ n where each S i +1 contains precisely one face of ∆ n more than S i (for 0 ≤ i < k ). Bythe vertical condition, the lift against S ⊆ T is completely determined by the liftsagainst the S i ⊆ S i +1 . But if ∆ m → S i +1 is the one face which belongs to S i +1 butnot S i , then its entire boundary lies in S i and we have a pullback diagram as follows: ∂ ∆ m S i ∆ m S i +1 . Since this diagram is both a square in the double category C and a pushout insimplicial sets, the lift against the map on the right is completely determined by thelift against the map on the left. (cid:3) In the remainder of this subsection we will try to answer the following question:suppose we have a map p : Y → X and we have chosen lifts against all boundaryinclusions ∂ ∆ m Y ∆ m X. pf i What conditions do these lifts f i have to satisfy in order for them to extend to alifting structure against C ?First of all, because any inclusion of sieves S ⊆ T ⊆ ∆ n can be seen as a composi-tion of pushouts of boundary inclusions, as in the previous lemma, we can solve anylifting problem of the form S YT X. p The first worry is that the decomposition of the inclusion S ⊆ T as a sequence ofinclusions where the next sieve contains one face more than the previous is in no wayunique, and it could be that the lift we construct depends on the decomposition. Asa matter of fact, it does not depend on this: imagine that we choose two differentdecompositions of the inclusion S ⊆ T and they determine lifts f and g , respectively.Now we can prove by induction on n ∈ N that f and g agree on all n -simplices, usingthat they agree on their boundaries in the induction step.The next worry is that these lifts need to satisfy both the horizontal and thevertical condition coming from C . The vertical condition is, in fact, automaticallysatisfied, because of the way we constructed the lifts and the fact that the way wedecompose the vertical maps in C is irrelevant.The horizontal condition can be split in two: because every map in ∆ is thecomposition of face and degeneracy maps, we only need to worry about squareswhere the map α is either a face or degeneracy map. In fact, the case where α is aface map is unproblematic. The reason is this: imagine that we have a decomposition S ⊆ S ⊆ S ⊆ . . . ⊆ S k ⊆ ∆ n and each S j +1 contains precisely one face more than S j . If we pull this back along d i : ∆ n − → ∆ n , we get either that d ∗ i S j = d ∗ i S j +1 if the face that gets added to S j in this step does not belong to d i , or that d ∗ i S j (cid:54) = d ∗ i S j +1 in case the face that getsadded to S j in this step does belong to d i . But in the latter case, d ∗ i S j +1 contains oneface more than d ∗ i S j , so if we ignore all the first cases we obtain a decomposition of d ∗ i S ⊆ d ∗ i S k . If we use this decomposition to compute the lift against d ∗ i S ⊆ d ∗ i S k , FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 83 then by pullback pasting ∂ ∆ m d ∗ i S j S j ∆ m d ∗ i S j +1 S j it is computed in exactly the same way as the lift against S ⊆ S k is computed onthe simplices which belong to the i th face.So the upshot of the discussion so far is that we only need to worry about thehorizontal condition for squares with α being a degeneracy map. Here, in view of thedecomposition, we can restrict attention to the situation where the map on the rightin the square is an inclusion S ⊆ T where T contains precisely one face more than S .In fact, we claim that we only need to worry about the situation where the map onthe right is a boundary inclusion, as in: s ∗ i ( ∂ ∆ n ) ∂ ∆ n Y ∆ n +1 ∆ n X ps i Indeed, assume that S ⊆ T ⊆ ∆ n is an inclusion where T contains precisely onemore face than S , which happens to be ∆ m → T ; also assume that we have somelifting problem of S ⊆ T against p , and s i : ∆ n +1 → ∆ n is some degeneracy. Usingthat pullbacks of monos along epis exist in the simplicial category, we can create adiagram as follows: s ∗ i S S YU ∂ ∆ m s ∗ i T T X ∆ m (cid:48) ∆ m ∆ n +1 ∆ n ∆ m (cid:48) ∆ m p Note that in the top cube, the left, right, front and back faces are squares in thedouble category C , and the right face is a pushout of simplicial sets. Therefore theleft face is a pushout as well and the horizontal condition for the back face of that cube is equivalent to the horizontal condition for the front of that cube. But the map∆ m (cid:48) → ∆ m is either the identity if i does not belong to the image of ∆ m → ∆ n , orsome degeneracy s j : ∆ m +1 → ∆ m if it does.This means that we can restrict attention to the following situation: s ∗ i ( ∂ ∆ n ) ∂ ∆ n Y ∆ n +1 ∆ n X u ps i f and let f be our favourite filler. Note that ∂ ∆ n = (cid:83) d nk and the pullback of d k along s i is d k if k < i , d i .d i if k = i and d k +1 if k > i ; in other words, s ∗ i ( ∂ ∆ n ) is ∆[ n + 1]with the interior and i th and ( i + 1)st faces missing. So to find the dotted filler inthe diagram above we first need to find the filler on the faces i and i + 1. So wepull back the left hand arrow along d i and d i +1 and choose our favourite filler, whichis, actually, f , because d i .s i = d i +1 .s i = 1. So we are left with the following fillingproblem: ∂ ∆ n +1 Y ∆ n +1 X. u ∪ f ∪ f p So what we need is that the chosen solution of this problem will be f · s i . (Note that f · s i will always be a solution. Indeed, we have f · s i · d k = ( u ∪ f ∪ f ) · d k for anyface d k : it is true on the faces that we added ( d i and d i +1 ), but also on the faces thatwere already there, because the original picture commutes.)So, to summarise our discussion, we have: Theorem 8.8.
The following notions of fibred structure are naturally isomorphic: • To assign to each p : Y → X all effective trivial Kan fibration structures onit. • To assign to each p : Y → X all systems of lifts of p against boundary inclu-sions such that if f is our chosen filler of ∂ ∆ n Y ∆ n X, y pfx then f.s i is our chosen solution of the problem ∂ ∆ n +1 Y ∆ n +1 X, y (cid:48) px.s i FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 85 where y (cid:48) is the composition s ∗ i ( ∂ ∆ n ) → ∂ ∆ n → Y on s ∗ i ( ∂ ∆ n ) and f on d i and d i +1 . Local character and classical correctness.
The characterisation given inTheorem 8.8 can both be used to show that our notion of an effective trivial Kanfibration is local and that it is classically correct. Let us first discuss local character.
Corollary 8.9.
The notion of an effective trivial Kan fibration is a local notion offibred structure.Proof.
As said, we use the characterisation in Theorem 8.8. So assume p : Y → X isa map of simplicial sets such that any pullback of it along a map x : ∆ n → X is aneffective trivial Kan fibration. If we have a lifting problem of the form ∂ ∆ n Y ∆ n X, y px then we can decompose it as follows ∂ ∆ n Y x Y ∆ n ∆ n X, y p x with a pullback on the right. Since the lifting problem on the right has a solution,by assumption, we also get a filler f for the outer rectangle. Since this definitionis forced, we only need to check the condition for the degeneracies. So then we arelooking at a situation like this: ∂ ∆ n +1 Y x · s i Y x Y ∆ n +1 ∆ n +1 ∆ n X y (cid:48) p s i x The lift against p is induced by the left hand square, but, by assumption, it is com-patible with the one coming from the composition of the two squares on the left,which is f.s i , as desired. (cid:3) It remains to check classical correctness, for which we use the following lemma,whose proof can be found in the appendix (see Proposition B.1).
Lemma 8.10.
A lifting problem of the form ∂ ∆ n X ∆ n has at most one degenerate solution (that is, if both x · σ and x · σ fill this trianglewith both σ i epis in ∆ different from the identity, then x · s = x · s ). Theorem 8.11.
Classically, any morphism which has the right lifting property withrespect to boundary inclusions ∂ ∆ n ⊆ ∆ n can be equipped with the structure of aneffective trivial Kan fibration.Proof. Suppose p : Y → X is a map for which we have a choice of fillers f i for everylifting problem of the form ∂ ∆ n Y ∆ n X. pf i (This uses the axiom of choice, depending on how one reads the assumption.) Then,using the Principle of Excluded Middle and the previous lemma, we may assume that f i is the unique degenerate solution, if it exists. Then the compatibility condition fromTheorem 8.8 is automatically satisfied, because it says that under certain conditionswe should choose the (unique) degenerate solution. (cid:3) Remark 8.12.
In view of the earlier work by Gambino, Henry, Sattler, Szumilo[GH19]; [GSS19]; [Hen19], one may wonder whether the previous result can be mademore constructive when degeneracy is decidable (in Y for instance). We fail to seehow it would, and for that reason the relationship with that work is far from clear tous. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 87 Simplicial sets as a symmetric Moore category
The purpose of this section is to show that the category of simplicial sets can beequipped with symmetric Moore structure. As we already mentioned in the intro-duction, the structure that we choose was already defined in the paper by Van denBerg & Garner [BG12] using the notion of simplicial Moore paths. However, since thenotion of a symmetric Moore structure that we work with in this paper is strongerthan that of a path object category as in [BG12], we have to verify some additionalequations. For checking that these hold, we use a new characterisation of the Moorepath functor M , namely, as a polynomial functor. For that reason we will first givea brief introduction to the theory of polynomial functors. Then we will define M asa polynomial functor and use this definition to check that it is a symmetric Moorestructure in the sense of this paper. Finally, we will prove that our new definition of M is equivalent to the one given in [BG12]. We will also isolate an interesting two-sided Moore structure on simplicial sets, which will give us effective left and rightfibrations.9.1. Polynomial yoga.
We start by recapping some general facts about polynomialfunctors. (Some useful references are: [Von15]; [AAG03]; [GK13].) Throughout thissection we will work in a category E which is locally cartesian closed and has finitecolimits. (We assume all this structure is chosen.) Definition 9.1. A polynomial in E is a morphism f : B → A in E . For reasons thatwill become clear soon, we will also write such morphisms as ( B a ) a ∈ A . We will referto A as the base and B a as a fibre . A morphism α of polynomials from f : B → A to g : D → C is a pair ( α + , α − ) consisting of a morphism α + : A → C and a morphism α − : A × C D → B making B A × C D DA C fα − gα + commute. So, basically, a morphism α from ( B a ) a ∈ A to ( D c ) c ∈ C ) consists of a map α + : A → C and a family of morphisms ( α − a : D α + ( a ) → B a ) a ∈ A . With this notation,composition of morphisms is given by( β + : C → E, β − c : F γ + ( c ) → D c ) ◦ ( α + : A → C, α − a : D α + ( a ) → B a ) =( β + .α + : A → E, α − a .β − α + ( a ) : F β + ( α + ( a )) → D α + ( a ) → B a ) . The result is a category which we will denote Poly( E ).In addition, let us write FEnd( E ) for the category of fibred endofunctors on E and fibred natural transformation between them. There is a functor P : Poly( E ) → FEnd( E ) sending a polynomial f : B → A to its associated polynomial functor P f : P f : E E /B E /A E . B ∗ Π f Σ A Written differently: P f ( X ) = (cid:88) a ∈ A (cid:89) b ∈ B a X = (cid:88) a ∈ A X B a = { ( a ∈ A, t : B a → X ) } . On morphisms α : ( f : B → A ) → ( g : D → C ), the functor P acts as follows: P ( α ) X : P f ( X ) → P g ( X ): ( a ∈ A, t : B a → X ) (cid:55)→ ( α + ( a ) ∈ C, t.α − a : D α + ( a ) → B a → X ) . Note that P ( α ) is a cartesian natural transformation (meaning: all naturalitysquares are pullbacks) if α − is iso.The following proposition will not be used in this paper, but explains the choiceof morphisms in the category Poly( E ). Proposition 9.2.
The functor P : Poly( E ) → FEnd( E ) is full and faithful.Proof. See [AAG03, Theorem 3.4]. (cid:3)
The category FEnd( E ) has finite limits and these are inherited by Poly( E ). Theterminal object is the polynomial 0 →
1. The product of ( f : B → A ) × ( g : D → C ) is[1 A × f, g × B ]: A × D + B × C → A × C. In other words, it has A × C as base, with fibre D a + B c over ( a, c ) ∈ A × C . Thepullback of δ : ( g : D → C ) → ( f : B → A ) and ϕ : ( h : F → E ) → ( f : B → A ) has C × A E as base, with the fibre P ( c,e ) over ( c, e ) being the pushout: B δ + ( c ) = B ϕ + ( e ) D c F e P ( c,e ) . δ − c ϕ − e In addition, the category FEnd( E ) carries a (non-symmetric) monoidal structuregiven by composition: F ⊗ G = F ◦ G . This is inherited by Poly( E ) as well: indeed,it carries a monoidal structure as follows:( B a ) a ∈ A ⊗ ( D c ) c ∈ C = { ( b ∈ B a , d ∈ D t ( b ) ) } ( a ∈ A,t : B a → C ) . Imagine that we have a morphism α : ( B a ) a ∈ A → ( B (cid:48) a (cid:48) ) a (cid:48) ∈ A (cid:48) and a morphism γ : ( D c ) c ∈ C → ( D (cid:48) c (cid:48) ) c (cid:48) ∈ C (cid:48) then α ⊗ γ = η with: η + ( a ∈ A, t : B a → C ) = ( α + ( a ) ∈ A (cid:48) , γ + .t.α − a : B (cid:48) α + ( a ) → B a → C → C (cid:48) ) η − ( a ∈ A,t : B a → C ) ( b (cid:48) ∈ B (cid:48) α + ( a ) , d (cid:48) ∈ D (cid:48) ( γ + .t.α − a )( b (cid:48) ) ) = ( α − a ( b (cid:48) ) ∈ B a , γ − t ( α − a ( b (cid:48) )) ( d (cid:48) ) ∈ D t ( α − a ( b (cid:48) )) )The monoidal unit I is 1 → Proposition 9.3.
The tensor ⊗ on Poly( E ) preserves pullbacks in both coordinates.Proof. Because pullbacks in functor categories are computed pointwise and polyno-mial functors preserve pullbacks. (cid:3)
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 89
We will also be interested in comonoids for this tensor: so, this consists of anobject M in Poly( E ) together with maps ε : M → I (the counit) and δ : M → M ⊗ M (the comultiplication) making the following MM M ⊗ M M M M δ M ⊗ εε ⊗ M M M ⊗ MM ⊗ M M ⊗ M ⊗ M δδ δ ⊗ M M ⊗ δ commute. We will refer to such objects as polynomials comonads .Every internal category in E determines such a comonad. Indeed, let C be aninternal category and cod: C → C be the codomain map. Then there is a counit: C C C cod ε − =id ε + =! and a comultiplication ( δ + , δ − ): cod → cod ⊗ cod with δ + : C → (cid:80) C ∈ C C cod − ( C )0 given by sending an object C in C to the pair ( C, λα ∈ cod − ( C ) . dom( α )), whilst( δ − ) C sends a pair ( α : D → C, β : E → D ) to α.β .This construction has a converse: indeed, one can show that every polynomialcomonad is induced in this way by an internal category (see [AU17]).Note that such a polynomial comonad is in particular a comonad (in the usualsense) on E and that the coalgebras for this comonad are precisely the internalpresheaves on C in E . Note also that such a polynomial comonad is automaticallystrong. Indeed, because a polynomial functor preserves pullbacks, we can think of astrength on P cod as a natural transformation α X : X × P cod (1) → P cod ( X ), or, in otherwords, as a map of polynomials (1: C → C ) ∼ = (1 → × (0 → C ) → (cod: C → C ). And there is a canonical such map: C C C C C α − =cod cod 1 cod α + =1 One readily checks this is indeed a strength and that with respect to this strengththe induced comonad is strong.9.2.
A simplicial poset of traversals.
Let us define an internal poset T in simpli-cial sets.The object of objects T has as its n -simplices the n -dimensional traversals . An n -dimensional traversal is a finite sequence of elements from [ n ] × { + , −} , that is, afunction θ : { , . . . , l } → [ n ] × { + , −} for some l ∈ N (including the empty traversal for l = 0). Perhaps a good way of drawing a traversal is as follows. An n -dimensionaltraversal is like a zigzag: • • p (cid:111) (cid:111) p (cid:47) (cid:47) • p (cid:47) (cid:47) • • p (cid:111) (cid:111) p (cid:47) (cid:47) • , a (possibly empty) sequence of edges pointing either to the left (-) or right (+), witha label p i ∈ [ n ]. The collection of such traversals is a simplicial set: the face map d i acts on such a traversal by removing all the edges labelled with i and relabelling theother edges (meaning: if an edge is labelled with j > i , replace that label by j − s i acts on such a traversal by duplicating edges labelled with i (withthe copies pointing in the same direction as the original edge) and labelling the firstcopy i + 1 and the second i in case the edge points to the right, and labelling the firstcopy i and the second i + 1 if the edge point to the left. Other edges are labelledaccordingly (meaning: if an edge was labelled j > i , then it now has the label j + 1).In general, the action by some α : [ m ] → [ n ] on such a traversal θ is given as follows:if the label of some edge is i , then replace it by α − ( i ) many edges pointing in thesame direction as the original edge, labelled by the elements of α − ( i ) in decreasingorder if the edge points to the right and in increasing order if the edge points to theleft. In short, θ · α is the unique map fitting into a pullback square { , . . . , l } [ m ] × { + , −}{ , . . . , k } [ n ] × { + , −} θ · αv α × θ with proj [ m ] . ( θ · α ): { , . . . , l } → [ m ] decreasing on those fibres v − ( i ) with θ ( i ) posi-tive, and increasing on those fibres v − ( i ) with θ ( i ) negative.A position in an n -dimensional traversal θ : { , . . . , l } → [ n ] × { + , −} is a choice ofone of the vertices: formally, it is an element p ∈ { , , . . . , l } . The elements of ( T ) n are pairs consisting of an n -dimensional traversal θ together with a position in thistraversal (a pointed traversal ). The action of α on the traversals is as before, while itacts on the choice of vertex as follows: if θ (cid:48) = θ · α , and v is some vertex in θ , then wechoose that vertex in θ (cid:48) which is the rightmost vertex in θ (cid:48) which is either the sourceor target of an edge which is a copy of an edge which was to the left of v (choosingthe leftmost vertex if no such edge exists).There are two maps cod , dom: T → T with cod being the obvious forgetful map(forgetting the choice of position), while dom removes the part of the traversal before the position. That means that we think of T as a simplicial poset with the finalsegment ordering ( θ ≤ θ if θ is a final segment of θ : in that case, there is aposition p in θ such that after that point we see θ , and p can be thought of as themorphism from θ to θ ).To see that this is an internal poset, note that there is a map id: T → T givenby choosing the position at the start of the traversal. Finally, we need a mapcomp: T × T T → T . FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 91
That is, we start with pointed traversals ( θ , p ) and ( θ , p ) such that θ is the finalsegment we obtain from θ by removing everything before position p . Then comptakes θ with position p (which is a position in θ , and, because θ is a final segmentof θ , in θ as well).In view of the correspondence between internal categories and polynomial comon-ads, this internal category induces a polynomial comonad, whose counit we denote s : cod → I and whose comultiplication we call Γ: cod → cod ⊗ cod. So if, M = P cod ,we have a strong comonad with counit s : M ⇒ M ⇒ M M . Remark 9.4.
Note that M ∼ = T . In fact, the object T was introduced as M Simplicial Moore paths. M has more structure: in fact, we have an internalcategory in Poly( E ). To see this, note that we can also equip T with the initialsegment ordering. In that case, we take the same codomain map, but now as domainmap we take dom ∗ : T → T which removes the part of the traversal after the chosenposition. In addition, we have a map id ∗ : T → T which chooses the endpoint of thegiven traversal as its chosen position, as well as an appropriate compositioncomp ∗ : T × T T → T . This means that M carries a second strong comonad structure with counit t : M ⇒ ∗ : M ⇒ M M .Note that with either ordering, the poset T has an initial object 0: 1 → T (theunique traversal of length 0), and with the final segment ordering, the map id ∗ : T → T points to the unique map from the initial traversal to the given traversal (andsimilarly for id and the initial segment ordering). This means that we also have amap r : I → cod given by: 1 1 T T r − =! cod r + =0 Note that because r − iso, the natural transformation induced by r is cartesian.At this point one readily checks all the axioms for a two-sided Moore structurewhich do not involve the multiplication µ . In fact, all of these follow simply fromthe fact that we are working in an internal category with an initial object 0 with theproperty that the only map C → s.r = t.r = 1 follows immediately from the fact that there isonly polynomial map I → I .(2) r is strong: α. (1 × r ) = r.p : X × → M X . In this we have to compare twomaps (1 → × (0 → → (cod: T → T ). In the forwards direction theyare both 0: 1 → T , while in the backwards direction they are both equal aswell, because they both have codomain 1.(3) Γ .r = rM.r , or: Γ .r = ( r ⊗ cod ) .r . We have (Γ .r ) + = Γ + .r + = (0 , λα : C → .C ), while ( r ⊗ cod .r ) + = (0 , λ : C → . backwards direction we again have to compare two maps which terminate in1: so these are again equal.(4) tM. Γ = r.t , or ( t ⊗ cod ) . Γ = r.t . Note ( t ⊗ cod . Γ) + ( C ) = dom(0 → C ) = 0, while ( r.t ) + ( C ) = 0. Also, ( r.t ) − C ( α : D →
0) =!: 0 → C and(( t ⊗ cod ) . Γ) − C ( α : D →
0) = comp(!: 0 → C, α : D → M t.
Γ = α. ( t, M !): cod → cod. Here ( t, M !): cod → (1: T → T ) is given by(1 T , id ∗ ), so that the right hand side is (1 , λα : D → C. !: 0 → C ). The lefthand side, however, is given by ( M t. Γ) + ( C ) = (1 cod ⊗ t )( C, λα : D → C.D ) = C , while ( M t. Γ) − C ( α : D → C ) = Γ − . (1 cod ⊗ t ) − ( α : D → C ) = comp( α : D → C, !: 0 → D ) =!: 0 → C , as desired.(6) Equations similar to those in (3–5) have to be verified for Γ ∗ as well: butsince also with the initial segment ordering, T has a strong initial object, thesame arguments will work.What is still needed, then, is to define µ X : M X t × sX M X → M X and to verify thatit satisfies all the expected equations.Using the formula for computing pullbacks of polynomials, we see that in order todefine µ we need maps T × T (cid:116) T × T T × T • T T × T T × , × cod] µ − cod µ + Note that the fibre over ( θ , θ ) of the map on the left is the collection of positionsin θ and θ , with the final position in θ identified with the initial position in θ . Sowhat we can do is define µ + ( θ , θ ) = θ ∗ θ , the concatenation of the two sequenceswith θ put before θ . Since the positions in θ ∗ θ are precisely the positions ineither θ or θ , with the final position in θ coinciding with the initial position in θ ,we have a pullback square T × T (cid:116) T × T T × T T T × T T . [cod × , × cod] µ ∗ cod µ + So we can choose µ − to be an isomorphism and µ will be a cartesian natural trans-formation.We will leave it to the reader to verify that µ is strong and combines with r, s, t to yield a category structure, which is both left and right cancellative. The mostdifficult axioms to check are the distributive laws and the sandwich equation, whichwe will discuss here in some detail, also because they were not part of [BG12]. Lemma 9.5.
The distributive law Γ .µ = µ. ( M µ.ν X . (Γ .p , α MX . ( p , M ! .p )) , Γ .p ): M X × X M X → M M X holds, as does the corresponding law for Γ ∗ . FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 93
Proof.
We only show the distributive law for Γ as the corresponding statement forΓ ∗ is proved similarly.We have to compare two mapscod × I cod → cod ⊗ cod . The left hand side goes via cod and in the positive direction sends ( θ , θ ) to ( θ ∗ θ , λp ∈ cod − ( θ ∗ θ ) . dom( p )), and in the negative direction sends a pair of positions p in θ ∗ θ and p in dom( p ) to the position corresponding to p in either θ or θ .Let us now try to decompose the right hand side. The map αM. ( p , M ! .p ): cod × I cod → cod ⊗ codis in the forwards direction a map T × T → P cod ( T ) which sends ( θ , θ ) to( θ , λp.θ ), while the backwards direction sends a pair of positions ( p ∈ θ , p ∈ θ )to p .Then the map M µ.ν. (Γ .p , α. ( p , M ! .p )) can be seen as a composition:cod × I cod (cod ⊗ cod) × cod ⊗ I (cod ⊗ cod)cod ⊗ (cod × I cod) cod ⊗ cod ∼ = 1 cod ⊗ µ where in the forwards directions these maps send ( θ , θ ) first to(( θ , λp ∈ cod − ( θ ) . dom( p )) , ( θ , λp ∈ cod − ( θ ) .θ ) , which gets rewritten to ( θ , λp ∈ cod − ( θ ) . (dom( p ) , θ ) , and then sent to( θ , λp ∈ cod − ( θ ) . dom( p ) ∗ θ ) . In the backwards direction it sends a pair of positions in p in θ and p in dom( p ) ∗ θ first to the pair p and the position corresponding to p either in dom( p ) or θ , andthen to the position corresponding to p in θ if it belongs to dom( p ) or to theposition corresponding to p in θ if it belongs to θ . In short, it sends p and p tothe position corresponding to p in either θ or θ .In the final step we look at the whole right hand side as a compositioncod × cod (cod ⊗ cod) × ⊗ cod (cod ⊗ cod)(cod × cod) ⊗ cod cod ⊗ cod . ∼ = In the positive direction this takes ( θ , θ ) first to(( θ , λp ∈ cod − ( θ ) . dom( p ) ∗ θ ) , ( θ , λp ∈ cod − ( θ ) . dom( p ))) , and then to ( θ ∗ θ , λp ∈ cod − ( θ ∗ θ ) . dom( p )), as before. In the backwards directionwe are given a pair consisting of a position p in θ ∗ θ and a position p in dom( p )and we start by making a case distinction on whether p lies in θ or θ . If it liesin θ , the pair gets mapped to the position p in θ . If it lies in θ , the pair gets mapped to the position corresponding to p in either θ or θ . So in either case itgets mapped to the position corresponding to p in either θ or θ , as before. (cid:3) Lemma 9.6.
The sandwich equation
M µ.ν. (Γ ∗ , Γ) = αM. (1 , M !): M → M M holds.Proof.
We have to compare two morphisms cod → cod ⊗ cod. The right hand sidecan be seen as a composition:cod cod × T T ⊗ cod cod ⊗ cod . ∼ = α ⊗ cod In the positive direction these maps are the diagonal T → T × T , a map T × T → T × T swapping the two arguments and a map T × T → P cod ( T ) sending ( θ , θ )to ( θ , λp.θ ). In short, in the positive direction this maps sends θ to ( θ, λp.θ ). Inthe negative direction, a pair of positions ( p, p (cid:48) ) in θ is sent to p (cid:48) .The left hand side can be seen as a compositioncod (cod ⊗ cod) × cod (cod ⊗ cod)cod ⊗ (cod × I cod) cod ⊗ cod . ∼ = 1 cod ⊗ µ In the positive direction this first sends θ to ( θ, dom ∗ , dom) and then it sends ( θ, t, t (cid:48) )to ( θ, λp.t ( p ) ∗ t (cid:48) ( p )). So the composition is ( θ, λp.θ ): the reason is that dom ∗ re-moves the part after the position, dom removes the part before the position, and *concatenates the results: so we just get the original traversal back. In the backwardsdirection a pair of positions ( p, p (cid:48) ) is first sent to the position corresponding to p (cid:48) ineither the part before or after p , and then it is sent to the corresponding position inthe whole traversal. In short, it is sent to p (cid:48) . (cid:3) This finishes the verification of the axioms for a two-sided Moore structure. Notethat all the proofs that we have given so far would still work if we restricted thetraversals in T to those which only move towards the right (that is, those traversals θ : { , . . . , l } → [ n ] × { + , −} for which θ ( i ) for any i ∈ { , . . . , l } is always of the form( k, +) for some k ∈ [ n ]). We will refer to the version of M that we get in this way as M + . Of course, similar remarks apply if we restrict the traversals to those that onlymove to the left; the version of M that we would have obtained in that way will bereferred to as M − . Theorem 9.7.
The endofunctors M , M + and M − equip the category of simplicialsets with three distinct two-sided Moore structures. It remains to check that M equips the category of simplicial sets with the structureof a symmetric Moore category. This means that we should be able to construct atwist map τ if we work with two different orientations. Indeed, in that case there is FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 95 a map of polynomials T • T T T τ − ∼ = cod τ + with τ + sending a traversal θ : { , . . . , l } → [ n ] × { + , −} to a traversal τ + ( θ ) with thesame length l and τ + ( θ )( i ) = τ ( θ ( l + 1 − i )), where τ ( k, +) = ( k, − ) and τ ( k, − ) =( k, +) for any k ∈ [ n ]. So τ + reverses the order and orientation of the traversal.Finally, τ − sends a position p in such a traversal to the position l − p . Note that τ − is an isomorphism and τ is a cartesian natural transformation.Most of the equations for τ are easy to verify, except perhaps for Γ ∗ = τ M.M τ. Γ .τ ,which is equivalent to Γ ∗ .τ = τ M.M τ. Γ, or Γ ∗ .τ = ( τ ⊗ τ ) . Γ. In the positive directionwe have that Γ ∗ .τ is the function sending a traversal θ to the pair where the firstcomponent is this traversal reversed, while the second component is the functionwhich takes a position in this reversed traversal and removes the part in this reversedtraversal after this position. In the backwards direction this takes a position in thereversed traversal and a position in the reversed traversal before the first position andproduces the position corresponding to the second position in the original traversal.The map τ ⊗ τ is the function which in the positive direction takes a pair ( θ, t : cod − → T ) to ( τ + ( θ ) , τ + ◦ t ◦ τ − ), so if t is the domain function (removing the part ofthe traversal before the given position), this agrees with the other function in thepositive direction. In the backwards direction, ( τ ⊗ τ ) . Γ takes a position in thereversed traversal and a position in the reversed traversal before the given position,first reverses both and then takes the position corresponding to the second positionin the original traversal, as before.This means that we have the following result, as promised:
Theorem 9.8.
The endofunctor M equips the category of simplicial sets with thestructure of a symmetric Moore category. In view of the results of Section 4.2, Theorem 9.7 implies that there are a priori six AWFSs on the category of simplicial sets. However, up to isomorphism, there areonly three, because • the twist map τ induces an isomorphism between the AWFS determined by( M + , Γ + , s ) and the one determined by ( M − , Γ ∗− , t ). • the twist map τ induces an isomorphism between the AWFS determined by( M + , Γ ∗ + , t ) and the one determined by ( M − , Γ − , s ). • the Moore structure determined by M is symmetric, so the AWFS determinedby ( M, Γ , s ) is isomorphic to the one determined by ( M, Γ ∗ , t ).In view of this we will make the following definition: Definition 9.9.
We will refer to the algebras for the monads of the three AWFSsabove as naive right fibrations , naive left fibrations and naive Kan fibrations , respec-tively. We will refer to the coalgebras for the comonad of the third AWFS as HDRs . Geometric realization of a traversal.
In this section we have defined M as the polynomial functor associated to cod: T → T . In [BG12], the Moore pathfunctor was defined differently (as a parametric right adjoint). The main goal of thissubsection is to show that the two descriptions are equivalent. To that purpose, wedefine the “geometric realization” of a traversal, a construction which can already befound in [BG12], and will also play an important role in the later sections. Definition 9.10.
For an element of the form ( k, ± ) ∈ [ n ] × { + , −} , let us define:( k, +) s = k + 1, ( k, +) t = k, ( k, − ) s = k, ( k, − ) t = k + 1. If θ is a n -dimensionaltraversal of length k , then we define its geometric realization (cid:98) θ to be the colimit ofthe diagram∆ n ∆ n . . . ∆ n ∆ n +1 ∆ n +1 ∆ n +1 d θ (1) s d θ (1) t d θ (2) s d θ (2) t d θ ( k ) s d θ ( k ) t in simplicial sets. In words: we turn an n -dimensional traversal into a simplicialset, by replacing its vertices by n -simplices and its edges by ( n + 1)-simplices, insuch a way that if an ( n + 1)-simplex comes from the i th edge, then the n -simplicescoming from the vertices connected by that edge are its θ ( i ) s -th and θ ( i ) t -th faces,respectively. Theorem 9.11.
The geometric realization (cid:98) θ of an n -dimensional traversal θ fits intoa pullback square (35) (cid:98) θ T ∆ n T . j θ k θ cod θ Proof.
We have to construct two maps j θ : (cid:98) θ → ∆ n and k θ : (cid:98) θ → T , which we will dousing that (cid:98) θ is a colimit. If θ ( i ) = ( k, ± ), let us write θ ( i ) = k . Then∆ n ∆ n . . . ∆ n ∆ n +1 ∆ n +1 ∆ n +1 ∆ nd θ (1) s d θ (1) t d θ (2) s d θ (2) t d θ ( k ) s d θ ( k ) t s θ (1) s θ (2) s θ ( k ) FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 97 commutes, so determines a map j θ : (cid:98) θ → ∆ n . In addition, we can construct coconewith vertex T :∆ n ∆ n . . . ∆ n ∆ n +1 ∆ n +1 ∆ n +1 T d θ (1) s ( θ, d θ (1) t d θ (2) s ( θ, d θ (2) t d θ ( k ) s ( θ, d θ ( k ) t ( θ,k ) a a a k In this diagram the n -simplices along the top correspond to positions p in θ , whichcorrespond to maps ( θ, p ): ∆ n → T . Furthermore, the i th ( n + 1)-simplex in thesecond row comes from the i th edge in θ and for this ( n + 1)-simplex we choose aposition in θ · s θ ( i ) : note that that in θ · s θ ( i ) the edge in question gets duplicated andthe position we choose is the one inbetween the two copies (we will refer to this as a“special position”). This determines maps a i : ∆ n +1 → T which make the diagramabove commute, and hence we obtain a map k θ : (cid:98) θ → T . Again using that (cid:98) θ is acolimit, it is not hard to see that with the resulting maps the square in the statementof the theorem commutes. It remains to show that it is a pullback.Imagine that we start with a map α : ∆ m → ∆ n and a position p in θ · α . Ourfirst task is to show there is some m -simplex in (cid:98) θ which gets mapped to α and p bythe maps we have just constructed. Let us partition the edges in θ · α by groupingtogether those edges which come from the same edge in θ : we will call these groups“blocks”. In other words, they are the fibres of the map v as in the pullback square { , . . . , l } [ m ] × { + , −}{ , . . . , k } [ n ] × { + , −} θ · αv α × θ determining θ · α . For the position p there are now two possibilities: the first is that p is the boundary between two blocks (that is, the edges to the right and left of p come from different edges in θ ; this is meant to include the case where p is one ofthe outer positions). In that case there is some position q in θ such that p is therestriction of q along α . Then we have the map ( θ, q ): ∆ n → T corresponding tothe position q , which we can also regard as an n -simplex in (cid:98) θ lying over the identityin ∆ n . Restricting this one along α , we get an element in (cid:98) θ of the form we want.The other case is that p belongs to the interior of one of the blocks with the edgesto the left and right mapping to the same edge with label i in θ . In that case we canuse that if α : [ m ] → [ n ] is some map in ∆, and α ( k ) = α ( k + 1) = i , then there isa map β : [ m ] → [ n + 1] such that α = s i .β and β ( k ) (cid:54) = β ( k + 1). (Indeed, we canput β ( j ) = α ( j ) if j ≤ k and β ( j ) = α ( j ) + 1 if j > k .) Then there is some specialposition q in θ · s i such that p is the restriction of q along β . This determines a map a j : ∆ n +1 → T , which we can also regard as an ( n + 1)-simplex in (cid:98) θ lying over s i in ∆ n . By restricting this ( n + 1)-simplex in (cid:98) θ along β , we get an m -simplex of the formwe want.It remains to show that the two maps (cid:98) θ → ∆ n and (cid:98) θ → T we constructed arejointly monic. Since every element in (cid:98) θ is a restriction of some a i , it suffices toprove the following statement: if a u and a v with u ≤ v are ( n + 1)-simplices in (cid:98) θ corresponding to ( θ · s i , p ) and ( θ · s j , q ), respectively, and α, β : ∆ m → ∆ n +1 are suchthat s i .α = s j .β and p · α = q · β , then a u · α = a v · β in (cid:98) θ .Let us first consider the case where i (cid:54) = j . For convenience, we will assume that i < j , and both edges i and j point to the right. Then we can take the followingpullback: ∆ n +2 ∆ n +1 ∆ n +1 ∆ n . s j +1 s i s j s i So there is some map γ such that α = s j +1 .γ and β = s i .γ and we obtain the equation p · s j +1 .γ = q · s i .γ . Note that both p · s j +1 and q · s i are distinct positions in thesame traversal and what the equation is saying is that they become identified afterrestricting along γ . The crucial observation is that this can only happen if γ removesthe edge to the right of p and the one to the left of q and everything else inbetween. Inparticular, γ factors through d i and d j +2 and we can write γ = d j +2 .d i .γ (cid:48) = d i .d j +1 .γ (cid:48) ,so that α = d i .γ (cid:48) and β = d j +1 .γ (cid:48) . Since γ (cid:48) must omit all the labels of edges betweeninbetween u and v , the following diagram commutes:∆ m ∆ n ∆ n . . . ∆ n +1 ∆ n +1 ∆ n +1 ∆ n +1 γ (cid:48) γ (cid:48) γ (cid:48) d ptu d psu +1 d ptu +1 d psu +2 d psv d ptu +3 Since the composite along the left is α and composite along the right is β , this showsthat a u · α = a v · β in (cid:98) θ , as desired. (There are other cases to be considered: differentdirections and i > j , but it all works out.)Let us now consider the case where i = j , but u < v . From s i .α = s i .β , it followsthat α − { i, i +1 } = β − { i, i +1 } = [ k, l ] for some k and l , whilst on inputs outside theinterval [ k, l ] the functions α and β are identical. In addition, we have the equation p · α = q · β , which implies that α must omit i and β must omit i + 1 (if, for simplicity,we assume that both edges point to the right; other cases are again similar). Thereason is that outside the i -blocks (that is, outside the pairs of consecutive edges in θ · s i of the form v − ( e ) with θ ( e ) = ( i, ± )) restricting along α and β acts in thesame way. But also on these i -blocks α and β act in very similar ways: both replacethem by strings of edges of length l − k with identical labels. The only difference isthat they may disagree on how to shift the special position. Hence to make the two FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 99 positions coincide one therefore has to shift the chosen position to the endpoints ofthe blocks and to eliminate the stuff inbetween. So α = d i .γ and β = d i +1 .δ andhence γ = δ . Since γ = δ must also omit every label inbetween the edges u and v , wecan again show (as in the previous case) that a u · α = a v · β in (cid:98) θ .Finally, if u = v , then s i .α = s i .β and p · α = p · β . The former equation againimplies that α − { i, i + 1 } = β − { i, i + 1 } = [ k, l ] for some k and l , whilst on inputsoutside the interval [ k, l ] the functions α and β are identical. But then the secondequation implies that also on the i -blocks α and β act in the same way and hence α and β also agree on the interval [ k, l ]. Hence α = β , and a u · α = a v · β . (cid:3) Corollary 9.12.
Geometric realization is part of a functor (cid:100) ( − ): (cid:90) ∆ T → (cid:98) ∆ . In fact, writing U : (cid:82) ∆ T → (cid:98) ∆ for the functor sending ( n, θ ) to ∆ n , we may regard the j θ from the previous theorem as the components of a cartesian natural transformation j : (cid:100) ( − ) → U .Proof. If θ is an n -dimensional traversal and α : ∆ m → ∆ n is some map in ∆, thenwe have two pullbacks (cid:100) θ · α (cid:98) θ T ∆ m ∆ n T . k θ · α (cid:98) αj θ · α j θ k θ cod α θ Therefore there exists a dotted arrow, turning the left hand square into a pullbackas well. (cid:3)
Corollary 9.13.
For the simplicial Moore path functor M we have ( M X ) n ∼ = (cid:88) θ ∈ T ( n ) Hom (cid:98) ∆ ( (cid:98) θ, X ) . Therefore the description given in this paper is equivalent to the one given in [BG12].Proof.
This is immediate from the following description of polynomial functors inpresheaf categories (see [MP00], for instance): if f : B → A is a morphism of presheavesover C , then P f ( X )( C ) = (cid:88) a ∈ A ( C ) Hom (cid:98) C ( B a , X ) , where B a is the pullback B a ByC A. fa (cid:3)
00 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
Remark 9.14.
A more abstract proof of the previous theorem and two corollariescan be found in the second author’s PhD thesis [Fab19]. It relies on proving thecategorical fact that when P : C op → S ets is a presheaf, U : E → S ets is its category ofelements, and F : E → [ C op , S ets ] is a functor with a cartesian natural transformation γ : F ⇒ y U ◦ ( − ) , then the square F ( e ) colim Fy C P ∼ = colim y U ◦ ( − ) induced by the cocones (horizontal) and γ (vertical) is a pullback square. This squarecorresponds to the square (35) in Theorem 9.11.For this reason we can think of the n -simplices in M X as pairs consisting of an n -dimensional traversal θ and a morphism π : (cid:98) θ → X of presheaves. We think of theseobjects as Moore paths in X . These Moore path generalise ordinary paths in X , thatis, maps I → X , in the following way.We have the 0-dimensional traversals ι + = (cid:104) (0 , +) (cid:105) and ι − = (cid:104) (0 , − ) (cid:105) which corre-spond to two global sections ι + , ι − : 1 ∼ = ∆ → T . Since I = ∆ is the geometric realization of both of these traversals, the previoustheorem tells us that we have pullback squares I T T . cod ι + /ι − Regarding these pullback squares as morphisms of polynomials, we obtain two moniccartesian natural transformations ι + , ι − : X I → M X . Furthermore, if we write s, t : X I → X for the maps induced by d : ∆ → ∆ and d : ∆ → ∆ , respectively, then we havethat the following diagrams serially commute:( − ) I MI ι + st ts ( − ) I MI ι − ts ts (So ι + preserves source and target, while ι − reverses them.) Remark 9.15.
Another way of seeing that the usual path object X I is a subobjectof M X is as follows. We can take the pullback of the square above along a map from
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 101 a representable ∆ n →
1: ∆ n × I I T ∆ n T . cod ι + /ι − Indeed, what this says is that ∆ n × ∆ is the geometric realisation of the traversals (cid:104) ( n, +) , ( n − , +) , . . . , (2 , +) , (1 , +) (cid:105) and (cid:104) (1 , − ) , (2 , − ) , . . . , ( n − , − ) , ( n, − ) (cid:105) . Indeed, this reflects the well-known decomposition of the “prism” ∆ n × ∆ as theunion of n many ( n + 1)-simplices; from our present point of view, this means that∆ n × ∆ occurs as the geometric realisation of these traversals. From this and thedescription of M in Corollary 9.13, one can also see that X I embeds in M X .
02 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
Hyperdeformation retracts in simplicial sets
In the previous section we have shown that the endofunctor M equips the categoryof simplicial sets with symmetric Moore structure. Consequently, the category ofsimplicial sets carries an AWFS, with the coalgebras for the comonad being calledthe HDRs and the algebras for the monad being called the naive Kan fibrations . Thepurpose of this section is to take a closer look at this AWFS.By definition, the naive Kan fibrations are generated by the large double categoryof HDRs. One important result in this section is that they are also generated by asmall (countable) double category of HDRs, and that the naive fibrations Kan forma local notion of fibred structure. It should be apparent from the proofs that similarresults would be true for naive left and right fibrations as well (see Definition 9.9).10.1.
HDRS are effective cofibrations.
Let us start by proving that HDRs areeffective cofibrations. As we have seen in Lemma 6.9, for this it suffices to prove thefollowing result:
Proposition 10.1.
The map r X : X → M X is always an effective cofibration.Proof.
In fact, since X ! (cid:47) (cid:47) r X (cid:15) (cid:15) r (cid:15) (cid:15) M X M ! (cid:47) (cid:47) M r is cartesian), it suffices to prove this statement for X = 1. In otherwords, we have to define a map ρ : T ∼ = M → Σ such that1 ! (cid:47) (cid:47) r (cid:15) (cid:15) (cid:62) (cid:15) (cid:15) M ρ (cid:47) (cid:47) Σis a pullback. We set ρ n ( θ ) = { e ⊆ { , , . . . , n } : e ∩ Im( θ ) = ∅} . (So we take those subsets e for which no i ∈ e occurs as ( i, +) or ( i, − ) in the image ofthe traversal θ . This happens precisely when the restriction of the traversal θ along e is the unique traversal of length 0.) This is easily seen to be correct. (cid:3) HDRs as internal presheaves.
In the sequel we will often have to provethat certain maps are HDRs. It turns out that for this purpose it is often convenientto use an equivalent description of the category of HDRs and morphisms of HDRs.Indeed, we have:
Theorem 10.2.
The category of HDRs in simplicial sets is equivalent to the categoryof internal presheaves on T . FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 103
Proof.
This is immediate from the fact that cod: HDR → M − Coalg is an equiva-lence (see Proposition 4.10) and the fact that M -coalgebras are equivalent to internalpresheaves over T . (cid:3) Let us unwind a bit more what this means and explain how one passes from HDRsto internal presheaves over T and back. If ( i : A → B, j, H ) is an HDR, then we cantranspose B → M B = (cid:88) θ ∈ T B cod − ( θ ) to a pair of maps d : B → T and ρ : B d × cod T T → B which satisfy the axioms foran internal presheaf over T . Conversely, suppose one is given an internal presheafover T , that is, a pair of maps d : B → T and ρ : B × T T → B satisfying the rightequations. The map H : B → M B is the transpose of ρ , while the inclusion i : A → B is the pullback A B T , i ! d so A is the fibre over the initial object in T . Finally, t.H : B → M B sends an elementin B over θ to its restriction along the unique map 0 → θ , and hence j does the same.Next, there are some constructions (pullback, pushout, and vertical composition)on HDRs that we would like to translate into the language of internal presheaves. Pullback:
Suppose d : B → T and ρ : B × T T → B is an internal presheaf and A is the fibre over 0 (that is, the pullback as above). If we are given a map a : A (cid:48) → A , then we obtain a new presheaf ( B (cid:48) , d (cid:48) , ρ (cid:48) ) by pullback as follows.First of all we take the pullback B (cid:48) BA (cid:48) A. bj (cid:48) ja If H : B → M B is the map induced by d, ρ , then H (cid:48) is the unique map filling B (cid:48) BM B (cid:48)
M BA (cid:48) × M M A (cid:48)
M A. H (cid:48) ( j (cid:48) ,M ! .H.b ) b HMj (cid:48) Mb Mjα Ma
04 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS So d (cid:48) : = M ! .H (cid:48) = M ! .H.b = d.b : B (cid:48) → T . Moreover, ρ (cid:48) will be the uniquedotted arrow filling B (cid:48) × T T B (cid:48) A (cid:48) B × T T B A. ρ (cid:48) j (cid:48) .π b × b j (cid:48) aρ j.π j Pushout:
Again, suppose d : B → T and ρ : B × T T → B is an internalpresheaf and A is the fibre over 0. If we are given a map a : A → A (cid:48) , we obtaina presheaf ( B (cid:48) , d (cid:48) , ρ (cid:48) ) by pushout, as follows. First, we take the pushout: A A (cid:48)
B B (cid:48) . ai i (cid:48) b If H : B → M B is the map induced by d, ρ , then H (cid:48) : B (cid:48) → M B (cid:48) is the uniquedotted arrow filling
A A (cid:48)
B B (cid:48) B (cid:48) M B M B (cid:48) . ai i (cid:48) i (cid:48) bH H (cid:48) rMb This means that d (cid:48) : B (cid:48) → T is the unique dotted arrow filling A A (cid:48)
B B (cid:48) T , ai i (cid:48) ! b d d (cid:48) whilst ρ (cid:48) is the unique map filling A A (cid:48) A × T T A (cid:48) × T T B × T T B (cid:48) × T T B B (cid:48) . a ∼ = ∼ = i (cid:48) a × i × i (cid:48) × b × ρ ρ (cid:48) b Vertical composition:
Suppose ( i : A → B, j , H ) and ( i : B → C, j , H )are HDRs coming from presheaf structures ( d , ρ ) and ( d , ρ ). Then thevertical composition is given by( i .i : A → C, j .j , µ. ( H , M i .H .j )) . So this means we have a function d : C → T , which is: d = M ! .µ. ( H , M i .H .j )= µ. ( M ! .H , M ! .M i .H .j )= µ. ( d , d .j )= d ∗ d .j . In addition, we need a morphism ρ : C × T T → C . Here the domain canalso be computed in two steps by taking the following two pullbacks: C × T T ∪ C × T T C T × T ∪ T × T T × T T T . [ π ,π ]( π ,d .j .π ) ∪ ( d ×
1) ( d ,d .j ) µ ∗ [cod × , × cod] ∗ cod Hence we can define ρ as [ ρ , i .ρ . ( j × C × T T ∪ C × T T → C .We finish this subsection with the proof that the category of HDRs contains a“generic” element. Proposition 10.3.
The triple (id ∗ : T → T , cod , (cid:91) comp) is an HDR, which is genericin the following sense: for any HDR ( i : A → B, j, H ) there exists a pullback ( i (cid:48) , j (cid:48) , H (cid:48) ) of the generic one together with a morphism of HDRs ( i (cid:48) , j (cid:48) , H (cid:48) ) → ( i, j, H ) which isan epimorphism on the level of presheaves.Proof. As a presheaf, the generic HDR is given by dom: T → T with comp: T × T T → T . Now imagine that we have some HDR, considered as a presheaf d : B → T
006 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS with ρ : B × T T → B . Then pulling back the generic HDR along d gives as presheafthe pullback B × T T B T T , p p d cod together with d (cid:48) = dom .p and ρ (cid:48) the unique filler of( B × T T ) × T T B × T T B T × T T T T . ρ (cid:48) p .π p × p p d comp cod .π cod One can interpret this presheaf as follows: the category of internal presheaves hasa forgetful functor to the slice category over T and this forgetful functor has a leftadjoint. The presheaf ( B × T T , d (cid:48) , ρ (cid:48) ) is the free presheaf on d : B → T . Thereforethere is an epic morphism of presheaves B × T T B T , ρd (cid:48) d namely the counit of the adjunction. (cid:3) A small double category of HDRs.
Our next goal is to show that thenaive Kan fibrations in simplicial sets are generated by a small double category. Wedo this by showing that the large double category of HDRs in simplicial sets containsa small double category such that a system of lifts against the small double categorycan always be extended in a unique way to a system of lifts again the entire doublecategory of HDRs.Let H be the following double category. • Objects are pairs ( n, θ ) with n ∈ N and θ an n -dimensional traversal. • There is a unique vertical map ( n , θ ) → ( n , θ ) if n = n and θ is a finalsegment of θ . • A horizontal map ( m, ψ ) → ( n, θ ) is a pair consisting of a map α : [ m ] → [ n ]together with an m -dimensional traversal σ such that ψ ∗ σ = θ · α . Theformula for horizontal composition is ( α, σ ) . ( β, τ ) = ( α.β, τ ∗ ( σ · β )). FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 107 • A square is any picture of the form( m, ψ ) ( n, θ )( m, ψ ) ( n, θ ) ( α,σ )( α,σ ) in which the horizontal arrows have the same label.Our first goal will be to argue that there is a double functor H → HDR( (cid:98) ∆ ) whichon the level of objects assigns to every ( n, θ ) the geometric realization of θ . Recallfrom the previous section that the geometric realization of θ is, by definition, thecolimit of the following diagram:∆ n ∆ n . . . ∆ n ∆ n +1 ∆ n +1 ∆ n +1 d θ (1) s d θ (1) t d θ (2) s d θ (2) t d θ ( k ) s d θ ( k ) t From now we will denote this colimit simply as θ , rather than as (cid:98) θ . Note that θ comes with two maps from ∆ n , corresponding to the outer maps ∆ n → ∆ n +1 in thisdiagram. We will refer to the map ∆ n → θ induced by the inclusion on the left as s θ and the map of the same shape induced by the inclusion on the right as t θ .In fact, s θ and t θ occur as pullbacks of id and id ∗ , as follows:∆ n T θ T ∆ n T s θ θ id j θ k θ cod θ ∆ n T θ T ∆ n T t θ θ id ∗ j θ k θ cod θ Indeed, using the notation of Corollary 9.12, we may regard s and t as cartesiannatural transformations U → (cid:100) ( − ) and sections of the cartesian natural transformation j : (cid:100) ( − ) → U . Also, since id ∗ is the generic HDR, we may regard t θ as an HDR.Moreover, the picture on the right as well as the naturality squares of the naturaltransformation t are cartesian morphisms of HDRs.A typical vertical morphism in H is of the form ψ → θ ∗ ψ . Such a morphism wecan equip with an HDR-structure, because the top square in∆ n ψθ θ ∗ ψ ∆ n ψ s ψ t θ ι ι j θ [ s ψ .j θ , s ψ
08 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS is a pushout. Note that because both squares are pullbacks as well, the diagram willbecome a bicartesian morphism of HDRs.This explains how we map vertical morphisms to HDRs. Let us now explain wherewe map the horizontal maps to. Note that the horizontal maps are of the form( α, σ ) which we can write as a composition ( α, (cid:104)(cid:105) ) . (1 , σ ), so we only need to explainwhere we map these composites to. However, the map (1 , σ ): ψ → ψ ∗ σ is just ι ,while (1 , α ) = (cid:98) α : ψ · α → ψ comes from the functoriality of geometric realization (seeCorollary 9.12; from now we will also simply write α ).To explain where we map the squares to, we use the same decomposition. Let usfirst look at a square coming from (1 , σ ):∆ n ψ ψ ∗ σθ θ ∗ ψ θ ∗ ψ ∗ σ ∆ n ψ ψ ∗ σ. t θ s ψ ι ι ι ι j θ ι s ψ s ψ ∗ σ ι The HDR-structures on the middle and right arrow are defined by pushout from theleft arrow: this automatically makes the right hand square a cocartesian morphismof HDRs, and, in particular, a morphism of HDRs. Note that, as before, all squaresin the diagram above are pullbacks as well, so the morphism is actually bicartesian.If the square comes from ( α, (cid:104)(cid:105) ), consider a double cube of the form: ψ · α ψ ∆ m ∆ n θ · α ∗ ψ · α θ ∗ ψθ · α θψ · α ψ ∆ m ∆ nαι ι s ψ · α αt θ · α s ψ α [ s ψ · α .j ψ · α ,
1] [ s ψ .j θ , αj θ · α ι ι t θ ααs ψ · α s ψ j θ Note that the left and right hand side of this double cube are cocartesian morphismsof HDRs, while the front is a cartesian morphism (since both come with a cartesianmorphism to the generic HDR). Since the bottom square is a pullback, the back is acartesian morphism of HDRs, by Beck-Chevalley.
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 109
This finishes the construction of a potential double functor H → HDR( (cid:98) ∆ ): the verification that it actually is a double functor turns out to be a lot of work and willkeep us occupied for the next couple of pages. Remark 10.4.
Note that all the squares in the image of this (potential) doublefunctor are cartesian morphisms of HDRs.
Lemma 10.5.
The potential double functor H → HDR( (cid:98) ∆ ) just constructed preserveshorizontal composition of morphisms.Proof. To prove that our double functor preserves horizontal composition it sufficesto check that ψ · α ψψ · α ∗ σ · α ψ ∗ σ αι ι α commutes. However, we have a commutative diagram∆ n ψ σψ ∗ σ ∆ n s σ t ψ ι j ψ ι j σ j ψ ∗ σ in which the square is a pushout, and by pulling it back along α : ∆ m → ∆ n we getthe commutativity of the previous square. (cid:3) Since our potential double functor trivially preserves horizontal identities and hori-zontal composition of squares, it remains to consider the vertical structure. Also herepreservation of identities and vertical composition of squares will be immediate oncewe show vertical composition of morphisms is preserved. For that, it is convenient touse an alternative construction of the HDR-structure on ψ → θ ∗ ψ , via the “genericinclusion of HDRs”. Indeed, consider the following picture: T T × T T × T T T × T T × T ∪ T × T T × T T T × T T × T ∗ id ∗ × × id π ι cod cod × ι π [cod × id , π × id We can give ι the structure of an HDR, by first pulling the generic structure on id ∗ back along π and then pushing the result forward along 1 × id. By pulling this HDR
10 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS back along the horizontal arrow at the centre of∆ n T × T ψ T × T ∆ n T × T θ,ψ ) s ψ × id j ψ × cod( θ,ψ ) we obtain ψ → θ ∗ ψ . The reason is that we can apply Beck-Chevalley to the topsquare in the diagram above and the HDR id ∗ × π .Let us translate the generic inclusion in presheaf language. First, the pullback ofthe generic HDR along π : T × T → T is T × T with d = dom .π and ρ = (comp . ( π .π , π ) , π .π ): ( T × T ) × T T → T × T . Then we need to push this forward along 1 × id: T × T → T × T , which results in: T × T T × T T × T T × T ∪ T × T T × idid ∗ × ι ! ι dom .π d So the generic inclusion of HDRs is T × T ∪ T × T with d = [dom .π , . !]. It alsocomes equipped with an action ρ , which we find as follows: T × T T × T ( T × T ) × T T ( T × T ) × T T ∪ T × T T × T T × T ∪ T × T ∗ × , id ∗ . . !) 1 × id ι ι ρι ρ ι Hence ρ = (comp . ( π .π , π ) , π .π ) ∪ π , ∗ : T × T → T . In terms of presheaves, the pullback of thegeneric HDR along π : T × T → T is d = dom .π : T × T → T , while ρ is the FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 111 unique map filling T × ( T × T T ) T × T T × T T × T T T T . π ρ ( π , cod .π .π ) 1 × cod π π comp cod Hence ρ = 1 × comp.To compute the pullback of the generic HDR along ∗ : T × T → T , we firstcompute the pullbacks T × T T T × T ∪ T × T T T × T T ι . (1 × id ∗ ) ∗ id ∗ µ ∗ [cod × , × cod] cod ∗ Hence the presheaf we are looking for is T × T ∪ T × T with d = dom .µ ∗ =[ ∗ . (dom .π , π ) , dom .π ], while ρ is the unique arrow filling( T × T ∪ T × T ) × T T T × T ∪ T × T T × T T × T T T T µ ∗ × ρ [cod × , × cod] .π [cod × , × cod] µ ∗ ∗ comp cod Lemma 10.6.
The vertical composition of ( d , ρ ) after ( d , ρ ) equals ( d , ρ ) .Proof. First of all, we have to prove that d = ∗ . ( d , d .j ): T × T ∪ T × T → T . This holds, because: ∗ . ( d , d .j ) .ι = ∗ . ( d .ι , d .j .ι )= ∗ . (dom .π , dom .π . (cod × id))= ∗ . (dom .π , dom . id .π )= ∗ . (dom .π , π )= d .ι
112 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS and ∗ . ( d , d .j ) .ι = ∗ . ( d .ι , d )= ∗ . (0 . ! , dom .π )= dom .π = d .ι . Writing C : = T × T ∪ T × T , this means that we can also think of the domainof ρ as the pullback C × T T ∪ C × T T C T × T ∪ T × T T × T T T . p q ( d ,d .j ) µ ∗ [cod × , × cod] ∗ cod where q : = [ π , π ] ,p : = ( π , d .j .π ) ∪ ( d × . Therefore the domain of ρ can be seen as the pushout of two pullbacks, the pullbackof d = [dom .π , . !]: C → T and cod as well as the pullback of d .j = dom .π . [cod × id ,
1] = [dom . id .π , dom .π ] = [ π , dom .π ]: C → T and cod. In these terms, ρ is the unique arrow making C × T T ∪ C × T T C T × T T T ρ [cod × , × cod] .q comp . ( µ ∗ .q,µ ∗ .p ) µ ∗ [cod × , × cod] ∗ cod commute.We have to prove ρ = [ ρ , i .ρ . ( j × µ ∗ and[cod × , × cod]. Note that the first summand C d × cod T T is isomorphic to[( T × T ) dom .π × cod T T ] ∪ [ T × T ] , FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 113 and in these terms we havecomp . ( µ ∗ .q, µ ∗ .p ) .ι =comp . ( µ ∗ .π , µ ∗ .ι . ( π , d .j .π )) =[comp . ( µ ∗ .ι .π , µ ∗ .ι . ( π , π .π ) , comp . ( µ ∗ .ι , µ ∗ .ι . (id ∗ . . ! , dom .π ))] =[ µ ∗ .ι . (comp . ( π .π , π ) , π .π ) , µ ∗ .ι ] = µ ∗ .ρ as well as [cod × , × cod] .q.ι =[cod × , × cod] .π =[cod × .π , × cod] =[(cod .π .π , π .π ) , × cod] =[cod × . (comp . ( π .π , π ) , π .π ) , × cod] =[cod × , × cod] .ρ . We now turn to the second summand, which we can write as C d .j × cod T T ∼ = ( T × T ) × T T ∪ ( T × T ) × T T ∼ = T × T ∪ T × ( T × T T ) , because d .j = [ π , dom .π ]. In these terms we havecomp . ( µ ∗ .q, µ ∗ .p ) .ι =comp . ( µ ∗ .π , µ ∗ .ι .d ×
1) =[comp . ( µ ∗ .ι . ( π , cod .π ) , µ ∗ .ι . (dom .π , π )) , comp . ( µ ∗ .ι . ( π , π .π ) , µ ∗ .ι . (0 . ! , π .π ))] =[ µ ∗ .ι . (cod .π , π ) , µ ∗ .ι . × comp] =which equals: µ ∗ .i .ρ . ( j ×
1) = µ ∗ .ι . (1 × comp) . ([cod × id , ×
1) = µ ∗ .ι . [(cod .π , comp . (id . cod .π , π )) , × comp] = µ ∗ .ι . [(cod .π , π ) , × comp] . In a similar fashion we have:[cod × , × cod] .i .ρ . ( j ×
1) =[cod × , × cod] .ι . (1 × comp) . ([cod × id , ×
1) =1 × cod . [(cod .π , π ) , × comp] =[(cod .π , cod .π ) , ( π , cod .π .π )] =[cod × , × cod] .π =[cod × , × cod] .q.ι . This finishes the proof. (cid:3)
14 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
Remark 10.7.
The previous lemma is equivalent to the distributive law in the form:Γ .µ = M µ. ( µM. (Γ .p , M r.p ) , µM. ( α. ( p , M ! .p ) , Γ .p )) . To prove that this reformulation is equivalent to the way we usually state the distribu-tive law, one would need an interchange law of the following form: suppose we haveelements α, β, γ, δ ∈ M M X with t ( α ) = s ( β ) , t ( γ ) = s ( δ ) , M t ( α ) = M s ( γ ) , M t ( β ) = M s ( γ ), then: M µ. ( µM ( α, β ) , µM ( γ, δ )) = µM. ( M µ. ( α, γ ) , M µ. ( β, δ )) . This interchange law may not hold in all Moore categories, but it can be shown tohold in the example at hand.In any case, to see the equivalence between the previous lemma and the reformula-tion of the distributive law, one would have to think about how one would prove thelatter, and for that we have to go back to the proof of Lemma 9.5. The descriptionof Γ .µ is still correct of course, but note that in the negative direction we can thinkof it as a map C × T T → C : C C × T T { ( θ ∈ T , t : cod − ( θ ) → T , p ∈ θ, p ∈ t ( p )) } T × T { ( θ, t : cod − ( θ ) → T ) } where the arrow along the bottom sends ( θ , θ ) to ( θ ∗ θ , d = d .µ ∗ : C → T ).With some effort one can recognise the map C × T T → C as ρ .Let us first look at the map µM. (Γ .p , M r.p ), which can be thought of as acomposite:cod × I cod (cod ⊗ cod) × I ⊗ cod (cod ⊗ cod)(cod × I cod) ⊗ cod cod ⊗ cod . ∼ = µ ⊗ cod In the positive direction this sends ( θ , θ ) first to (( θ , λp. dom( p ) , ( θ , λp. θ ∗ θ , d = [dom .π , . !]: C → T ). Then in the negativedirection we have a map C d × cod T T → C which with some effort one can recogniseas ρ .Let us now have a look at µM. ( α. ( p , M ! .p ) , Γ .p ). We can think of it as amorphismcod × I cod (cod ⊗ cod) × I ⊗ cod (cod ⊗ cod)(cod × I cod) ⊗ cod cod ⊗ cod . ∼ = µ ⊗ cod In the positive direction this sends a pair ( θ , θ ) first to (( θ , λp.θ ) , ( θ , λp. dom( p ))which then gets sent to what is essentially ( θ ∗ θ , d .j = [ π , dom .π ]: C → T ). FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 115
This means that in the negative direction we should have a map C d .j × cod T T → C :again with some effort one can recognise this as i .ρ . ( j × × I cod (cod ⊗ cod) × cod ⊗ I (cod ⊗ cod)cod ⊗ (cod × I cod) cod ⊗ cod . ∼ = 1 cod ⊗ µ In the positive direction this will be a map sending ( θ , θ ) to ( θ ∗ θ , ∗ . ( d , j .d ): C → T ); in the backwards direction this is a map C × T T ∪ C × T T → C which is[ ρ , i .ρ . ( j × Proposition 10.8.
There is a double functor H → HDR( (cid:98) ∆ ) which on the level ofobjects assigns to every traversal its geometric realization.Proof. As said, it remains to check that vertical composition of morphisms is pre-served. We start by pulling back the vertical composition from the previous lemmaalong ( θ, ψ ): ∆ n → T × T and obtain the following picture:∆ n T × T ψ T × T θ ∗ ψ T × T ∪ T × T ψ T × T ∆ n T × T t θ ∗ ψ t ψ × id ∗ ι . (1 × id ∗ ) ι ι [cod × id , × cod( θ,ψ ) Since vertical composition is preserved by pullback (see part 1) and we have a com-mutative triangle of HDRs on the right, the same is true on the left. Since pushforward also preserves vertical composition, we can push this triangle forward along s ψ : ∆ n → ψ to see that our double functor does indeed preserve vertical composi-tion. (cid:3) This lengthy verification was only the first step towards proving the main resultof this subsection, which is:
Theorem 10.9.
The following notions of fibred structure are isomorphic: • Having the right lifting property against the large double category of HDRs insimplicial sets (that is, to be a naive Kan fibration). • Having the right lifting property against the small double category H .
16 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
More precisely, the morphism of notions of fibred structure induced by the doublefunctor from H to the large double category of HDRs in simplicial sets is an isomor-phism.Proof. First, we explain how one can construct a morphism of notions of fibred struc-ture in the other direction. So imagine that we have a map p : Y → X which has theright lifting property against H and we are given a lifting problem A YB X gi pfl where ( i : A → B, j, H ) is an HDR.In the proof of the existence of a generic HDR, we have seen that for any HDR( i, j, H ) there are two morphisms of HDRs, as in T B A T B × T T B, id ∗ (1 , id ∗ .d ) jd ip ρ where the one on the left is cartesian. So if b ∈ B n , we can pull the middle HDR backalong b : ∆ n → B and we obtain a commutative diagram in which the left square is amorphism of HDRs, as follows:∆ n A Yθ B X, t θ j.b i g pπL b f if d ( b ) = θ . By assumption, we have a dotted filler L b , as indicated. Now we put l ( b ): = L b .s θ .Let us first check that this defines a natural transformation l : B → Y . If weconsider b · α for some α : ∆ m → ∆ n , then we have a picture as follows:∆ m ∆ n A Yθ · α θ B X, αt θ · α t θ j.b i g p (cid:98) αL b · α π L b f Hence l ( b · α ) = L b · α .s θ · α = L b . (cid:98) α.s θ · α = L b .s θ .α = l ( b ) .α, which shows that l is indeed a natural transformation.Let us now check that l is fills the original square. Because π.s θ = b , we have p.l ( b ) = p.L b .s θ = f.π.s θ = f ( b ) , FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 117 hence l makes the lower triangle commute. Also, if b = i ( a ) for some a ∈ A n , then d ( b ) = (cid:104)(cid:105) (the empty traversal), and l.i ( a ) = L b .s (cid:104)(cid:105) = L b .t (cid:104)(cid:105) = g.j ( b ) = g.j.i ( a ) = g ( a ) . It remains to check that these lifts satisfy both the horizontal and vertical com-patibility conditions. We start by looking at the horizontal one. Imagine we have acommutative diagram of the form: A (cid:48) A YB (cid:48)
B X αi (cid:48) i pβl (cid:48) l where the square on the left is a morphism of HDRs and b (cid:48) ∈ B (cid:48) n . Then this fits intoa larger picture: A (cid:48) A YB (cid:48) B ∆ n B (cid:48) B XB (cid:48) × T T B × T T θ αi (cid:48) i pj (cid:48) β (1 , id ∗ .d (cid:48) ) jb (cid:48) β ( b (cid:48) ) t θ ββ × ρ (cid:48) ρ (1 , id ∗ .d ) with d (cid:48) = d.β . So from left to right we obtain a lift π : θ → Y and l (cid:48) ( b (cid:48) ) = π.s θ . Butbecause the front face of the cube is a cartesian morphism of HDRs (as one easilychecks), this is also l ( β ( b (cid:48) )). This shows the horizontal compatibility condition.For checking the vertical compatibility condition, imagine that we have a liftingproblem of the form: A YBC X i i pi l l with the HDR i the vertical composition of i and i , l and l the lifts induced by i and i , respectively, and c ∈ C n . The aim is to show that for the induced lift l : C → Y induced by i we have l ( c ) = l ( c ). Using the formulas for the vertical composition of
18 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
HDRs (in the language of presheaves) we obtain a commutative diagram, as follows:∆ n C B Aθ C × T T B × T T B ∆ n Cθ ∗ θ C × T T ∪ C × T T Cθ C × T T ct θ j (1 , id ∗ .d .j ) (1 , id ∗ .d ) j i ι j × ι ρ i s θ t θ (1 , id .d .j ) j ρ =[ ρ ,i .ρ . ( j × ι ι ρ (1 , id ∗ .d ) Here d ( c ) = θ ∗ θ with θ = ( d .j )( c ) and θ = d ( c ). Note that l ( c ) = π .s θ ∗ θ where π : θ ∗ θ → Y is the lift coming from the lifting structure of p against H .However, π .s θ ∗ θ can be computed in two steps: we first compute the lift π : θ → Y .Then we use π .s θ to compute π : θ → Y and then we have π .s θ ∗ θ = π .s θ . Butit follows from the diagram that π .s θ = l ( j .c ) and s θ .π = l ( c ): so l ( c ) = l ( c ),as desired.We have constructed two operations between two notions of fibred structure: nowit remains to show that they are mutually inverse. It is easy to see that if we startfrom a map having the RLP against all HDRs, then only remember the lifts againstthe vertical maps in H and then use the operation defined above to compute a liftagainst a general HDR, we return at our starting point. The reason is simply thatthe left hand square in ∆ n A Yθ B X, t θ j.b i g pπL f l is a morphism of HDRs, so we must have L = l.π . So if b = π.s θ , then L.s θ = l.π.s θ = l ( b ).The converse turns out to be a lot harder. Suppose that we start with a map p having the right lifting property against all maps in H , and that we are given alifting problem of p against a vertical map from H . Now we can find a solution intwo different ways: first, we can use the lifting structure of p directly. Alternatively,we can use that vertical maps in H are HDRs and use this to find a lift, followingthe procedure explained above. The question is: are both solutions necessarily thesame? We claim that the answer is yes.To prove the claim, it suffices to check that the lifts against traversals of length1 are identical. The reason is that any inclusion of traversals can be written as the FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 119 vertical composition of inclusions where each next traversal has length one longerthan the previous. And if we have a traversal of the form σ → (cid:104) ( i, ± ) (cid:105) ∗ σ , then thereis a bicartesian square in H of the form: (cid:104)(cid:105) σ (cid:104) ( i, ± ) (cid:105) (cid:104) ( i, ± ) (cid:105) ∗ σ So the lift against the map on the right is completely determined by the lift againstthe map on the left.So let us imagine that we have an inclusion of traversals of the form (cid:104)(cid:105) → (cid:104) ( i, +) (cid:105) (we will only look at the positive case, for simplicity). Then its geometric realisationis: ∆ n T ∆ n +1 T ∆ n T d i (cid:104) ( i, +) (cid:105) id ∗ s i u cod (cid:104) ( i, +) (cid:105) where u = k (cid:104) i, + (cid:105) picks out the traversal (cid:104) ( i, +) (cid:105) · s i = (cid:104) ( i + 1 , +) , ( i, +) (cid:105) with thespecial position (so the position in the middle). Note that this means that facemaps d i : ∆ n → ∆ n +1 are HDRs: let us see what its HDR-structure is in presheaflanguage. First of all, we have d = dom .u = (cid:104) ( i, +) (cid:105) : ∆ n +1 → T . Secondly, we haveto determine ρ : ∆ n +1 × T T → ∆ n +1 . But note that the domain of ρ also arises asthe centre left object in ∆ n +1 T ∆ n +2 T ∆ n +1 T d i d id ∗ vs i cod d = (cid:104) ( i, +) (cid:105) where v chooses (cid:104) ( i, +) (cid:105) · s i with the special position; in other words, it is isomorphicto ∆ n +2 . This means that ρ is the unique map filling∆ n +2 ∆ n +1 ∆ n T × T T T T . ( u · s i ,v ) ρ s i .s i q u s i ( (cid:104) i, + (cid:105) )comp cod20 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS Note that u · s i is (cid:104) ( i + 2 , +) , ( i + 1 , +) , ( i, +) (cid:105) with the position between the firstand second element, and therefore q is (cid:104) ( i + 2 , +) , ( i + 1 , +) , ( i, +) (cid:105) with the positionbetween the second and third elements. We conclude that ρ must be s i +1 .What we have to prove, then, is that if we have a lifting problem of the form∆ n Y ∆ n +1 X d i p where we think of the arrow on the left as an HDR coming from the inclusion oftraversals (cid:104)(cid:105) → (cid:104) ( i, +) (cid:105) , then the lift coming from the fact that it is an HDR coincideswith the one coming from the fact that p has the right lifting property against H .The former lift is computed by choosing an arbitrary element α ∈ ∆ n +1 and pullingthe map on the left back along s i .α . But because ∆ n +1 is representable it sufficesto do this for α = 1, which means that what we have to prove is that if we have apicture as follows: ∆ n +1 ∆ n Y ∆ n +2 ∆ n +1 X s i d i d i ps i +1 l l then for the lifts coming from the fact that p has the right lifting property against H , we have l = l .d i +1 . For that it suffices to prove l .s i +1 = l . (Just to be clear,the left and centre vertical arrows are to be thought of as HDRs coming from theinclusion of traversals (cid:104)(cid:105) → (cid:104) ( i, +) (cid:105) in dimensions n + 1 and n , respectively.)To that end, note that we have a commutative diagram of the form:∆ n +1 ∆ n Y (cid:104) ( i, +) (cid:105)(cid:104) ( i + 1 , +) , ( i, +) (cid:105) (cid:104) ( i, +) (cid:105) X ∆ n +1 ∆ ns i d i d i pw ι l w [ s i +1 ,s i ] l s i l s i Here we have pulled d i back along s i and then decomposed it vertically. This meansthat the map w = (cid:98) s i is the unique map making (cid:104) ( i + 1 , +) , ( i, +) (cid:105) (cid:104) ( i, +) (cid:105) T ∆ n +1 ∆ n T w [ s i +1 ,s i ] k (cid:104) ( i +1 , +) , ( i, +) (cid:105) s i k (cid:104) ( i, +) (cid:105) cod s i (cid:104) ( i, +) (cid:105) FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 121 commute. Here k (cid:104) ( i +1 , +) , ( i, +) (cid:105) = [ z , z ] with z choosing the position between thefirst and second element in (cid:104) ( i + 2 , +) , ( i + 1 , +) , ( i, +) (cid:105) and z choosing the positionbetween the second and third. Therefore w = [ s i , s i +1 ] and w = s i +1 . This meansthat the top left square in the previous diagram coincides with the left hand squarein the diagram before that. Therefore also the lifts l and l in both diagram mustcoincide and since they are compatible with l , we deduce l .s i +1 = l , as desired.This completes the proof for the positive case: the negative case is similar. (cid:3) Naive Kan fibrations in simplicial sets.
From the previous theorem, weget two new descriptions of the naive Kan fibrations. Both start by observing that theentire lifting structure against H is already determined by a subclass of the verticalmaps. First of all, we can consider those inclusions (cid:104)(cid:105) → θ with empty domain: anyother lift is completely determined by these, because (cid:104)(cid:105) ψθ θ ∗ ψ t θ s ψ ι ι is a cocartesian square. Therefore the lift against the arrow on the right has to bethe pushout of the lift against the arrow of the right. So one can equivalently definea naive Kan fibration structure in terms of lifts against arrows of the form (cid:104)(cid:105) → θ . Ifone does so, the horizontal compatibility condition for maps of the form (1 , σ ) dropsout and we are left with the horizontal compatibility condition for maps of the form( α, (cid:104)(cid:105) ). In other words, we have: Corollary 10.10.
The following notions of fibred structure are equivalent: • To assign to a map p : Y → X all its naive Kan fibration structures. • To assign to a map p : Y → X a function which given any n -dimensionaltraversal θ and commutative square ∆ n Yθ X t θ p chooses a lift θ → Y . Moreover, these chosen lifts should satisfy two condi-tions:(i) If α : ∆ m → ∆ n , then the chosen lifts ∆ m ∆ n Yθ · α θ X αt θ · α t θ p are compatible.
22 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS (ii) If θ = θ ∗ θ , then the chosen lift l : θ → Y can be computed in two steps:we can first compute the lift ∆ n Yθ θ X t θ pl and then compute the chosen lift l for ∆ n θ Yθ θ X t θ s θ l pl and push this forward to obtain a map l : θ → Y (so l = [ l , l ] ). Using Corollary 9.13, the second bullet in the previous condition can be seen as anexternalisation of conditions (1) – (3) in the definition of a naive fibration structure on p (see Definition 4.22). So the previous proposition says that there is a isomorphism ofnotions of fibred structure from maps carrying a naive fibration structure satisfyingconditions (1) – (4) to those which only satisfy (1) – (3). It turns out that thisisomorphism is just the forgetful map: Corollary 10.11. If p : Y → X is a map in simplicial sets, then any map L : Y × X M X → M Y automatically satisfies condition (4) for a naive fibration structure as soon as it sat-isfies conditions (1) – (3).Proof.
Let us call a map L : Y × X M X → M Y a weak naive fibration structure ifit only satisfies conditions (1) – (3). Then we know that there is an isomorphismbetween the transport structures on p and the weak naive fibration structures on p obtained by the operations studied in this section. Let us see how we get a transportstructure from a weak naive fibration structure in this way. We start by extendingthe weak naive fibration structure to a right lifting structure against H , which canthen be extended to all HDRs. This can be used to find the transport structure t bysolving the problem Y YY × X M X X ,r.p ) ps.p using that the map on the left is an HDR, via δ p = ( α. ( p , M ! .p ) , Γ .p ): Y × X M X → M ( Y × X M X ) ∼ = M Y × MX M M X.
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 123
This means that on an arbitrary element ( y, ( θ, π : θ → X )): ∆ n → Y × X M X , thevalue t ( y, ( θ, π )) is the solution to a lifting problem:∆ n Y Yθ Y × X M X X yt θ ,r.p ) pz s.p where z is the transpose of δ p ( y, ( θ, π )). But that means that s.p .z is the transposeof M s.p . ( α. ( p , M ! .p ) , Γ .p )( y, ( θ, π )) = ( θ, π ); in other words, s.p .z = π and theinduced lift is L ( y, ( θ, π )). Therefore the induced transport structure is defined by t ( y, ( θ, π )): = s.L ( y, ( θ, π )).So the upshot is that L (cid:55)→ s.L is the isomorphism of notion of fibred structurefrom the weak naive fibration structures to the transport structures. But we haveseen in Proposition 4.24 that this also defines an isomorphism of notions of fibredstructure between ordinary naive fibration structures and transport structures. Weconclude that every weak naive fibration structure already satisfies condition (4). (cid:3) Remark 10.12.
We believe that the previous corollary can also be shown directly.Very roughly, the reason is the following. One can think of Γ as being built from pathcomposition and degeneracies, and since any weak naive fibration structure L is inparticular a morphism of simplicial sets, it will automatically respect degeneracies.So if L respects path composition, it must also respect Γ.If p is a naive Kan fibration, its lifting structure against H is also completelydetermined by its lifts against the inclusions of traversals of the form (cid:104)(cid:105) → (cid:104) ( i, ± ) (cid:105) .Indeed, we already used and explained this in the proof of Theorem 10.9: any verticalmap in H is a vertical composition of inclusions of traversals where the next traversalhas length one more than the previous and each such inclusion is a pushout of oneof the form (cid:104)(cid:105) → (cid:104) ( i, ± ) (cid:105) . In the remainder of this section we will determine whichcompatibility conditions the lifts against these maps have to satisfy in order to extendto a (unique) lifting structure against H . This description will also allow us to provethat the notion of a being a naive Kan fibration is a local notion of fibred structure.If we are given the lifts against the maps (cid:104)(cid:105) → (cid:104) ( i, ± ) (cid:105) and we extend them tothe entire double category H in the manner described above, then both the verticalcompatibility condition as well as horizontal compatibility condition for maps of form(1 , σ ) are automatically satisfied. So we only need to ensure the horizontal compati-bility condition for maps of the form ( α, (cid:104)(cid:105) ). To ensure that, we only need to considersquares where the horizontal map α is either a face or degeneracy maps and thevertical maps on the right is one of the form (cid:104)(cid:105) → (cid:104) ( i, ± ) (cid:105) .We obtain the following cases:
24 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS (i) For the face maps, we have compatibility conditions for the case k < i (left)and the case k > i (right):∆ n − ∆ n ∆ n ∆ n +1 ∆ n − ∆ nd k d ( i − , ± ) t d ( i, ± ) t s i − d k s i d k ∆ n − ∆ n ∆ n ∆ n +1 ∆ n − ∆ nd k d ( i, ± ) t d ( i, ± ) t s i d k +1 s i d k What we mean here is that we have a horizontal compatibility condition forthe top squares in both diagrams below, in that if p : Y → X is a naive Kanfibration, and we have lifting problem as in∆ n − ∆ n Y ∆ n ∆ n +1 X ∆ n − ∆ nd k d ( i − , ± ) t d ( i, ± ) t ps i − d k s i d k ∆ n − ∆ n Y ∆ n ∆ n +1 X ∆ n − ∆ nd k d ( i, ± ) t d ( i, ± ) t ps i d k +1 s i d k then the dotted lifts have to be compatible. (Note that there is also a case k = i , but it is trivially satisfied, because in that case we get the identityinclusion (cid:104)(cid:105) → (cid:104)(cid:105) on the left.)(ii) For the degeneracy maps, we have compatibility condition for the case k < i (left) and k > i (right):∆ n − ∆ n ∆ n ∆ n +1 ∆ n − ∆ ns k d ( i +1 , ± ) t d ( i, ± ) t s i +1 s k s i s k ∆ n − ∆ n ∆ n ∆ n +1 ∆ n − ∆ ns k d ( i, ± ) t d ( i, ± ) t s i s k +1 s i s k as in (i).(iii) Pulling back (cid:104) ( i, ± ) (cid:105) along s i is a rather special case, which we split in botha positive and negative case (on the left and right, respectively).∆ n +1 ∆ n ∆ n +1 ∆ n +2 ∆ n +2 ∆ n +2 ∪ ∆ n +1 ∆ n +2 ∆ n +1 ∆ n +1 ∆ ns i d i d i d i +1 d i +1 s i +1 ι ι [ s i ,s i +1 ][ s i +1 ,s i ] s i s i ∆ n +1 ∆ n ∆ n +1 ∆ n +2 ∆ n +2 ∆ n +2 ∪ ∆ n +1 ∆ n +2 ∆ n +1 ∆ n +1 ∆ ns i d i +2 d i +1 d i +1 d i +1 s i ι ι [ s i +1 ,s i ][ s i ,s i +1 ] s i s i FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 125
Therefore we obtain compatibility conditions as follows:(a) In the positive case for:∆ n +1 ∆ n Y ∆ n +2 ∆ n +1 X ∆ n +1 ∆ ns i d i d i y ps i +1 s i l s i xl s i ∆ n +1 ∆ n Y ∆ n +2 ∆ n +1 X ∆ n +1 ∆ nd i +1 l .d i +1 d i y ps i s i +1 l s i xl s i The diagram on the left expresses a compatibility condition similar to theprevious ones (even in that the top left square is a morphism of HDRs:see the proof of Theorem 10.9). The one on the right is different, becausethere is no map ∆ n +1 → ∆ n making the top left hand square commute.Note that the diagram on the left implies that l .d i +1 = l .s i +1 .d i +1 = l ,so the reference to l in the diagram on the right can be eliminated.(b) In the negative case we have similar compatibility conditions:∆ n +1 ∆ n Y ∆ n +2 ∆ n +1 X ∆ n +1 ∆ ns i d i +2 d i +1 y ps i s i +1 l s i xl s i ∆ n +1 ∆ n Y ∆ n +2 ∆ n +1 X ∆ n +1 ∆ nd i +1 l .d i +1 = l d i +1 y ps i +1 s i l s i xl s i Remark 10.13.
Note that in this notion of fibred structure we do not just chooselifts for each commutative square with some d i : ∆ n → ∆ n +1 on the right: we are alsogiven as input a retraction of d i (which has to be either s i − or s i : ∆ n +1 → ∆ n ). Soalthough the lifting problem in no way refers to this retraction, the lifting structuremay choose different solutions if d i comes equipped with a different retraction. Also,the compatibility condition is formulated not for the d i as such, but for the d i togetherwith a choice of retraction: indeed, the compatibility condition takes this choice intoaccount in a crucial way.From this characterisation we immediately get: Corollary 10.14.
In the category of simplicial sets being a naive Kan fibration is alocal notion of fibred structure.
Another thing which this definition of a naive Kan fibration makes clear is that thetraversals with positive and negative orientation live in parallel universes and thereare no compatibility conditions relating the two. Indeed, to equip a map with thestructure of a naive Kan fibration means equipping it with the structure of a naive
26 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS right fibration and with the structure of a naive left fibration, with no requirementson how these two structures should relate. Put differently, we have:
Corollary 10.15.
In the category of notions of fibred structure, the notion of beinga naive Kan fibration is the categorical product of the notion of being a naive rightfibration and the notion of being a naive left fibration.
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 127
Mould squares in simplicial sets
In this and the next section we will study effective Kan fibrations in simplicial sets.By definition, they are those maps which have the right lifting property against thelarge triple category of mould squares, with mould squares coming from the Moorestructure M . The main aim of this section is to show that there is a small triplecategory of mould squares which generates the same class. In the next section wewill use this to show that the effective Kan fibrations in simplicial sets form a localnotion of fibred structure. Remark 11.1.
The attentive reader will notice that results similar to the ones wederive here hold for the mould squares coming from the two other Moore structureson simplicial sets (see Theorem 9.7 and Definition 9.9). We will refer to the mapshaving the right lifting property against the triple category of mould squares comingfrom ( M + , Γ + , s ) as the effective right fibrations and the maps having the right liftingproperty against the triple category of mould squares coming from ( M + , Γ ∗ + , t ) as the effective left fibrations . Implicitly we will show that these are also generated bysuitable small triple categories of mould squares.11.1. Small mould squares.
We will define a triple category M as follows. • Objects are triples ( n, S, θ ), usually just written (
S, θ ), consisting of a naturalnumber n , a cofibrant sieve S ⊆ ∆ n and an n -dimensional traversal θ . • There is a unique horizontal morphism ( S , θ ) → ( S , θ ) if S = S and θ is a final segment of θ . • There is a unique vertical morphism ( S , θ ) → ( S , θ ) if θ = θ and S ⊆ S is an inclusion of cofibrant sieves. • Perpendicular morphisms ( T ⊆ ∆ m , ψ ) → ( S ⊆ ∆ n , θ ) are pairs ( α, σ ) with α : ∆ m → ∆ n and σ an m -dimensional traversal such that α ∗ S = T and ψ ∗ σ = θ · α . Perpendicular composition is given by ( α, σ ) . ( β, τ ) = ( α.β, τ ∗ ( σ · β )),as before. • The triple category is codiscrete in the xy -plane in that whenever pairs ofhorizontal and vertical arrows fit together as in( S , θ ) ( S , θ )( S , θ ) ( S , θ ) , then this is the boundary of a unique square. We will refer to such a squareas a small mould square . • In the yz - and xz -plane squares exist as soon as the perpendicular arrowshave the same label ( α, σ ) (and the domains and codomains match up), andany two such which are “parallel” (have identical boundaries) are identical. • The triple category is codiscrete in the third dimension, in that any potentialboundary of a cube contains a unique cube filling it.
28 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
Proposition 11.2.
There is a triple functor M → MSq( (cid:98) ∆) from the triple category M to the large triple category of mould squares in simplicial sets.Proof. Perhaps it is good to remind the reader of the structure of the large triplecategory of mould squares in simplicial sets: • The objects are simplicial sets. • The horizontal morphisms are HDRs. • The vertical morphisms are cofibrations. • The perpendicular morphisms are arbitrary maps of simplicial sets. • The squares in the xy -plane are mould squares (morphisms of HDRs whichare cartesian over a cofibration). • The squares in the xz -plane are morphisms of HDRs. • The squares in yz -plane are morphisms of cofibrations (that is, pullbacks). • The cubes are pullback squares of HDRs (of a mould square along an arbitrarymorphism of HDRs).The idea is to send the object ( n, S, θ ) to the pullback θ · S in simplicial sets: θ · S θS ∆ n . j θ In the x -direction we send ( S, θ ) → ( S, θ ) to the HDR we obtain by pullback: S ∆ n θ · S θ θ · S θ . Note that both squares become cartesian squares of HDRs. Because pullback pre-serves composition of HDRs, this operation preserves composition in the x -direction.Similarly, in the y -direction we send ( S , θ ) → ( S , θ ) to the cofibration we obtain bypullback, as follows: θ · S θ · S θS S ∆ n . From this it immediately follows that the squares in the xy -plane are sent to mouldsquares in simplicial sets. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 129
The next step should be that squares on the left are sent to morphisms of HDRsand squares on the right to morphisms of cofibrations:(
T, ψ ) (
S, θ )( T, ψ (cid:48) ) (
S, θ (cid:48) ) ( α,σ )( α,σ ) ( T, ψ ) (
S, θ )( T (cid:48) , ψ ) ( S (cid:48) , θ ) ( α,σ )( α,σ ) We will again split this up in the case where α = 1 and the case where σ = (cid:104)(cid:105) . If α = 1, then S = T , S (cid:48) = T (cid:48) , θ = ψ ∗ σ and ψ (cid:48) = τ ∗ ψ and θ (cid:48) = τ ∗ θ for some traversal τ . In this case, the square on the left is sent to pullback of the right hand square∆ n ψ ψ ∗ στ τ ∗ ψ τ ∗ ψ ∗ σ. t τ s ψ s ψ ∗ σ ι ι ι ι ι along S → ∆ n . Since pullback preserves bicartesian morphisms of HDRs (Beck-Chevalley!), the result is a bicartesian morphism of HDRs. In addition, since theouter rectangle and the right hand square in ψ · S θ · S Sψ · S (cid:48) θ · S (cid:48) S (cid:48) are pullbacks, the square on the left is as well. Therefore the right hand square inthe earlier diagram will be sent to a morphism of cofibrations when α = 1.Let us now consider the case σ = (cid:104)(cid:105) ; now ψ = θ · α and ψ (cid:48) = θ (cid:48) · α . Then we needto show that the front face of the bottom cube in∆ m ∆ n T Sψ θψ · T θ · Sψ (cid:48) θ (cid:48) ψ (cid:48) · T θ (cid:48) · S α
30 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS is a morphism of HDRs. But since all the other faces in this cube (besides the leftand right one) are cartesian morphisms of HDRs, so must be the front face. Similarly,we need to show that the top of the left cube in θ · S θ · S (cid:48) θψ · T ψ · T (cid:48) ψS S (cid:48) ∆ n T T (cid:48) ∆ m α is a pullback. But it is not hard to see that all faces in both cubes must be pullbacks.From the fact that the the squares in the yz -plane are pullbacks, it follows fromCorollary 4.18 that the cubes are sent to pullback squares of HDRs. (cid:3) Remark 11.3.
Note that it follows from the proof that the morphisms of HDRs thatoccur as images of squares in the xy -plane are cartesian. Theorem 11.4.
The following notions of fibred structure in simplicial sets are iso-morphic: • Having the right lifting property against the large triple category of mouldsquares (that is, to be an effective Kan fibration). • To have the right lifting property against the small triple category M .More precisely, the morphism of notions of fibred structure induced by the triple func-tor from M to the large triple category of mould squares in simplicial sets is anisomorphism.Proof. For reasons that will become clear later, we will first prove that both notions offibred structure are equivalent if we ignore the vertical condition on both sides (so onboth sides we have lifts satisfying only the horizontal and perpendicular conditions).After we have done that, we will show that the equivalence restricts to one where onboth sides the vertical condition is satisfied as well.So suppose p : Y → X has the right lifting property against the small mould squaressatisfying the horizontal and perpendicular conditions, and assume we are given alifting problem of the form C D YA B X pi l where the square on the left is a mould square. We wish to find a map l : B → Y making everyting commute; for that, assume that we are given some b ∈ B n . Let us FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 131 write ( i : A → B, j, H ) for the HDR-structure on i . As we have seen in the previoussection, we can construct a morphism of HDRs∆ n Aθ B iπ with π.s θ = b and j ( b ) = θ . By pulling back the mould square along this morphismof HDRs, we obtain a picture as follows: C D YS θ · SA B X ∆ n θ pi Since the mould square at the front of the cube belongs to M , the picture induces amap L b : θ → Y making everything commute. We put l ( b ): = L b .s θ , as in the previoussection. At this point we need to verify a number of things: that this defines a naturaltransformation B → Y , that this map fills the square and is compatible with the map A → Y that we were given. Also, we need to verify that if we choose these lifts forthe mould squares, then together these lifts satisfy the horizontal and perpendicularcompatibility conditions. All of these things are just very minor extensions of resultsproved in the previous section, so we will omit the proofs here.We will now show that the operation we have just defined and the one inducedby the triple functor from the previous proposition are each other’s inverses. Onecomposite is clearly the identity: if we are given a map p : Y → X which has the rightlifting property against mould squares satisfying the horizontal and perpendicularconditions, restrict it to M and then extend it all mould squares in the mannerdescribed above, then we end up where we started. The reason is simply that thecube in the diagram above is a “mould cube” (belongs to the large triple category ofmould squares).The converse is the hard bit: so imagine that we have a map p : Y → X which hasthe right lifting property against the small mould squares satisfying the horizontaland perpendicular conditions. This means that if we have a lifting problem of theform: ( S , θ ) ( S , θ ) Y ( S , θ ) ( S , θ ) X, p we can solve in two different ways. First of all, we can use the lifting structure of p directly; but we can also observe that the square on the left is a large mould square
32 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS and use the procedure outlined above to find the lift. The task is to show that bothlifts are the same. Again, we argue as in the previous section, by first observing thatwe can reduce this problem to the situation where θ = (cid:104)(cid:105) and θ = (cid:104) i, ±(cid:105) . Indeed,we can write the mould square on the left as a horizontal composition of small mouldsquares where the traversal on the right has one entry more than the one on the left.Moreover, we have a mould cube( S , θ ) ( S , (cid:104) i, ±(cid:105) ∗ θ )( S , (cid:104)(cid:105) ) ( S , (cid:104) i, ±(cid:105) )( S , θ ) ( S , (cid:104) i, ±(cid:105) ∗ θ )( S , (cid:104)(cid:105) ) ( S , (cid:104) i, ±(cid:105) ) (1 ,θ ) (1 ,θ )(1 ,θ ) (1 ,θ ) in which the top and bottom faces are cocartesian. Therefore the lift against the backis completely determined by that the lift against the front face. In fact, we can takeit one step further: if α : ∆ m → ∆ n ∈ S , then( S , (cid:104)(cid:105) ) ( S , (cid:104) i, ±(cid:105) )( α ∗ S , (cid:104)(cid:105) ) ( α ∗ S , (cid:104) i, ±(cid:105) )( S , (cid:104)(cid:105) ) ( S , (cid:104) i, ±(cid:105) )(∆ m , (cid:104)(cid:105) ) (∆ m , (cid:104) i, ±(cid:105) ) ( α,
1) ( α, α,
1) ( α, is a mould cube as well. This means that the lift against the back face is completelydetermined by the lifts against the front faces if we let α range over S . In short, weonly have to compare lifts against small mould squares of the form:( S, (cid:104)(cid:105) ) ( S , (cid:104) i, ±(cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, ±(cid:105) ) X. p But this can be argued for just as in the previous section, so we again omit the proof.It remains to check that this equivalence of notions of fibred structure restricts toone where the vertical condition is satisfied one both sides. In fact, we only need toshow that if p : Y → X comes equipped with lifts against the small mould squares(satisfying the vertical condition as well), and we extend this to all mould squaresin the manner explained above, then the lifts against all the mould squares satisfy FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 133 the vertical condition. Before we do that, we make the important point that in thisextension of the lifting structure to all mould squares as in
C D YS θ · SA B X ∆ n θ pi lπ L b we must have l.π = L b . Indeed, this follows from the fact that the two ways ofcomputing of lifts against small mould squares coincide.So imagine we have a lifting problem of the form: E F YC DA B X pi in which the two squares on the left are mould squares. Imagine that we have chosensome b ∈ B n and constructed our morphism of HDRs from ∆ n → θ to i , as before.Then we can pull this vertical composition of mould squares back along this morphism
34 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS of HDRs, and pull that back along some arbitrary morphism α ∈ S , as follows: E F YS θ · S α ∗ S ( θ · α ) · ( α ∗ S ) C DS θ · S ∆ m θ · αA B X ∆ n θ ∆ m θ · α p What this amounts to is saying that our lift L b : θ → Y can be computed by firstcomputing the map L b · S : θ · S → Y . But that map is completely determined by themaps L b · α : θ · α → Y with α ranging over S . From this the vertical condition for thelarge mould squares at the back follows. (cid:3) Effective Kan fibrations in terms of “filling”. If p : Y → X has the rightlifting property against M , then it comes equipped with a choice of lifts against everysmall mould square, where these lifts satisfy several compatibility conditions. Becauseof these compatibility conditions some of the lifts are completely determined by thechoices we made for other lifts. What we can do is try to identify a suitable subclassand express the compatibility conditions purely in terms of lifts against elements inthis smaller subclass. This is the game we have played already a number of times. Forthe small mould squares, we will take this to the limit in the next section, but herewe can already note that the lifts general mould squares are completely determinedby those of the form: ( S, (cid:104)(cid:105) ) ( S, θ )(∆ n , (cid:104)(cid:105) ) (∆ n , θ ) . FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 135 (Here ∆ n stands for the maximal sieve on ∆ n .) Indeed, we have already implicitlyargued for this in the previous proof. Indeed, if we have a small mould square of theform ( S , ψ ) ( S , θ ∗ ψ )( S , ψ ) ( S , θ ∗ ψ )then there is a mould cube of the form( S , ψ ) ( S , θ ∗ ψ )( S , (cid:104)(cid:105) ) ( S , θ )( S , ψ ) ( S , θ ∗ ψ )( S , (cid:104)(cid:105) ) ( S , θ ) (1 ,ψ ) (1 ,ψ )(1 ,ψ ) (1 ,ψ ) in which the top and bottom faces are cocartesian: therefore the lifts against theback in completely determined by the lift against the front. Furthermore, if we havea mould square as in the front of this mould cube, it occurs at the back of a mouldcube ( S , (cid:104)(cid:105) ) ( S , θ )( α ∗ S , (cid:104)(cid:105) ) ( α ∗ S , θ )( S , (cid:104)(cid:105) ) ( S , θ )(∆ m , (cid:104)(cid:105) ) (∆ m , θ ) ( α,
1) ( α, α,
1) ( α, where α : ∆ m → ∆ n ∈ S . Since the ( α, m , θ ) → ( S , θ ) collectively cover( S , θ ), any compatible system of lifts against the front squares (while α ranges over S ) descends to a unique lift against the front. Let us call the lift against the backthat we obtain in this way the induced lift . Then we have the following result, whoseproof we omit because it is a variation on a type of argument we have already seen anumber of times. Proposition 11.5.
The following notions of fibred structure are equivalent: • To assign to each map p : Y → X all its effective Kan fibration structures. • To assign to each map p : Y → X all functions which given a natural num-ber n ∈ N , a cofibrant sieve S ⊆ ∆ n , an n -dimensional traversal θ and a
36 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS commutative square ∆ n ∪ θ · S Yθ X m [ t θ ,i θ ] pn choose a filler θ → Y . Moreover, these chosen fillers should satisfy the fol-lowing three compatibility conditions:(1) for each α : ∆ m → ∆ n the choice of filler for the composed square ∆ m ∪ ( θ · α ) · ( α ∗ S ) ∆ n ∪ θ · S Yθ · α θ X m pn is the composition of θ · α → θ with the chosen filler θ → Y for the righthand square.(2) if θ = θ ∗ θ , then the chosen filling for ∆ n ∪ θ · S Yθ X [ y , [ m ,m ]] pn coincides with the one we obtain in the following manner. One can firstcompute the filler for the composed square ∆ n ∪ θ · S ∆ n ∪ θ · S Yθ θ X, [ y ,m ] [ y , [ m ,m ]] pn from which we get an element y : ∆ n → Y by precomposition with thesource map s θ : ∆ n → θ . Then we can compute the filler for the square ∆ n ∪ θ · S Yθ θ X. [ y ,m ] pn By amalgamating the two maps θ i → Y we just constructed, we obtainanother map θ → Y , which is the one which should coincide with thefiller for the original square. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 137 (3) if S ⊆ S ⊆ ∆ n then the chosen filler for ∆ n ∪ θ · S Y ∆ n ∪ θ · S θ X a pb coincides with the one we obtain by first taking the induced lift ∆ n ∪ θ · S → Y of p against a and then the chosen lift θ → Y of p against b . Using the description of M as a polynomial functor (see Corollary 9.13), one canalso express the second item in the previous corollary as follows: to equip a map p : Y → X with the structure of an effective Kan fibration means choosing a map L : (cid:88) ( y,θ ) ∈ Y × X MX (cid:88) σ ∈ Σ M Y σ ( y,θ ) → M Y such that:(1) L exhibits ( t, M p ) as an effective trivial fibration, that is, L fills M Y M Y (cid:80) ( y,θ ) ∈ Y × X MX (cid:80) σ ∈ Σ M Y σ ( y,θ ) Y × X M X t,Mp ) L and is an algebra map (for the AWFS coming from the dominance).(2) L ( y, θ ∗ θ , ( σ, ρ ∗ ρ )) = L ( y, θ , ( σ, ρ )) ∗ L ( s.L ( y, θ , ( σ, ρ )) , θ , ( σ, ρ )) forall generalised elements y ∈ Y , θ , θ ∈ M X , σ ∈ Σ and ρ , ρ ∈ M Y σ .From this we immediately obtain: Corollary 11.6. If p : Y → X is an effective Kan fibration, then ( t, M p ): M Y → Y × X M X is an effective trivial fibration.
The fact that there are cartesian natural transformations ι + , ι − : X I → M X as inthe previous section, means that we have pullback squares of the form Y I M YY × X X I Y × X M X. ( s/t,p I ) ( t,Mp ) And since effective trivial fibrations are stable under pullback, we can deduce:
38 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
Corollary 11.7. If p : Y → X is an effective Kan fibration, then ( s/t, p I ): Y I → Y × X X I are effective trivial fibrations. Therefore effective Kan fibrations are uniform Kanfibrations in the sense of [GS17]. Remark 11.8.
This means that we can obtain our notion of an effective Kan fibrationfrom Gambino and Sattler’s notion of a uniform Kan fibration by making the followingchanges:(1) We replace the path object X I by the simplicial Moore path object M X .(2) We demand that the lifts behave well with respect to concatenation of Moorepaths, which implies that lifts of general simplicial Moore paths are deter-mined by those of length 1.(3) We work with a double category of cofibrations, rather than a category ofcofibrations, giving rise to an additional vertical requirement reflecting com-position of cofibrations.Changes (1) and (2) are motivated by our desire to make the notion of an effectiveKan fibration local. A good way to think about this is as follows: an n -simplex in Y I corresponds to a prism ∆ n × I → Y . By regarding ∆ n × I as the geometric realisationof a traversal, that is, as the union of n -many ( n + 1)-simplices (see Remark 9.15),and by saying that the lifts ∆ n × I → Y should be determined by what happens onthe Moore paths of lenght one, we say that lifts are determined by wat happens onthese ( n + 1)-simplices. In doing so, we obtain a local definition, as we will prove inthe next section.The third requirement is not strictly necessary for that purpose, but, as we willsee in future work, it will have the consequence that the effective trivial fibrationsand the effective Kan fibrations interact nicely. FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 139
Horn squares
The purpose of this section is to show that our notion of an effective Kan fibrationin simplicial sets is both local and classically correct. By the latter we mean that,in a classical metatheory, a map can be equipped with the structure of an effectiveKan fibration precisely when it has the right lifting property against horns (the tra-ditional notion of a Kan fibration). To prove both these statements we will use acharacterisation of the effective Kan fibrations in terms of what we will call hornsquares.12.1.
Effective Kan fibrations in terms of horn squares.
Recall that the smallmould squares are the squares in the yz -plane in the triple category M (see previoussection). Definition 12.1.
A small mould square will be called a one-step mould square ifin the horizontal direction the length of the traversal increases by one and in thevertical direction the sieve increases by one m -simplex which was not yet present,but all whose faces were. Among these one-step squares are the horn squares ( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) ( i, ± ) (cid:105) )(∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) ( i, ± ) (cid:105) )which start from the empty traversal in the horizontal direction and end with themaximal sieve in the vertical direction.The reason for the name horn square is the following: a lifting problem for p against a horn square ( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) ( i, ± ) (cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) ( i, ± ) (cid:105) ) X p is equivalent to a lifting problem for p against the map from the inscribed pushoutof the left hand square to its bottom right corner:∆ n ∪ ∆ n +1 · ∂ ∆ n Y ∆ n +1 X. p Here ∆ n +1 · ∂ ∆ n = Λ n +1 i,i +1 ∪ ( d i ∩ d i +1 ), that is, ∆ n +1 with the interior and the i th and( i + 1)st faces missing. Therefore in the previous square the map on left is the horninclusion Λ n +1 i +1 → ∆ n +1 in the positive case and the horn inclusion Λ n +1 i → ∆ n +1 inthe negative case. Note that it follows from this that effective Kan fibration have theright lifting property against horns, so are Kan fibrations in the usual sense.
40 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
Remark 12.2.
To make notation less cluttered, we will, from now on, write thetraversal (cid:104) ( i, ± ) (cid:105) as (cid:104) i, ±(cid:105) . Lemma 12.3.
If a map is an effective Kan fibration in that it has the right liftingproperty against the triple category M of small mould squares, then this structure iscompletely determined by its lifts against the horn squares.Proof. The proof combines three reductions, each of which we have already seenbefore.First of all, any inclusion S ⊆ T of sieves can be written as a sequence S = S ⊆ S ⊆ S ⊆ . . . ⊆ S n = T where at each point S i +1 is obtained from S i by addingone new m -simplex whose boundary was already present in S i . Therefore any smallmould square can be decomposed into a grid A , A , A , . . . A n, A , A , A , . . . A n, A , A , A , . . . A n, . . . . . . . . . . . . . . .A ,m A ,m A ,m . . . A n,m of one-step mould squares. Therefore the lifts against the one-step mould squaresdetermine everything.Secondly, as we have already seen in the proof of Theorem 11.4 the lifts againstthe one-step mould squares are determined by the one-step mould squares startingfrom the empty traversal. The reason, once again, is that there is a mould cube( S , θ ) ( S , (cid:104) i, ±(cid:105) ∗ θ )( S , (cid:104)(cid:105) ) ( S , (cid:104) i, ±(cid:105) )( S , θ ) ( S , (cid:104) i, ±(cid:105) ∗ θ )( S , (cid:104)(cid:105) ) ( S , (cid:104) i, ±(cid:105) ) (1 ,θ ) (1 ,θ )(1 ,θ ) (1 ,θ ) whose bottom face is a pushout.Thirdly, suppose we have a one-step mould square starting from an empty traversalin the horizontal direction and suppose that in the vertical direction we have the FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 141 inclusion S ⊆ T , where α : ∆ m → ∆ n is the m -simplex that has been added to S toobtain T . Then ∂ ∆ m S ∆ m T α is bicartesian and we get a small mould cube of the form:( S, (cid:104)(cid:105) ) ( S, (cid:104) i, ±(cid:105) )( ∂ ∆ m , (cid:104)(cid:105) ) ( ∂ ∆ m , (cid:104) i, ±(cid:105) · α )( T, (cid:104)(cid:105) ) ( T, (cid:104) i, ±(cid:105) )(∆ m , (cid:104)(cid:105) ) (∆ m , (cid:104) i, ±(cid:105) · α ) . ( α, (cid:104)(cid:105) ) ( α, (cid:104)(cid:105) )( α, (cid:104)(cid:105) ) ( α, (cid:104)(cid:105) ) Since in this cube the right hand face is a pushout, the lift against the back face isdetermined by its front face. But because α is monic, the front face is either a hornsquare or trivial in the horizontal direction, depending on whether i is in the imageof α or not. (cid:3) The remainder of this section will almost exclusively be devoted to answering thefollowing question: suppose we are given a map p together with chosen lifts againstthe horn squares. Which conditions do these lifts have to satisfy in order for them toextend to a (necessarily unique) effective Kan fibration structure on p ?Throughout the following discussion we assume that we have fixed a map p : Y → X together with a choice of lifts (or pushforwards) with respect to all horn squares. Wehave seen how these lifts can be extended first to lifts against one-step mould squaresand then to lifts against small mould squares. The worry we have to address iswhether the reductions in Lemma 12.3 determine these lifts in unambiguous manner.For the one-step mould squares there is no such problem, but for the small mouldsquares this is far from clear. Indeed, imagine that we have a lifting problem as
42 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS follows: A , A , A , . . . A n, Y p A , A , A , . . . A n, A , A , A , . . . A n, . . . . . . . . . . . . . . .A ,m A ,m A ,m . . . A n,m X, in which all the little squares are one-step mould squares. We have unambiguouspushforwards for every square ( A i,j , A i +1 ,j , A j +1 ,i , A i +1 ,j +1 ) in that we have chosenfor every pair of maps A i +1 ,j → Y and A i,j +1 → Y over X and under A i,j anextension to a map A i +1 ,j +1 → Y . Then for every map A ,m → Y we can build apush forward to a map A n,m → Y by repeatedly taking our chosen push forwards forthe one-step mould squares. The first worry is that we can travel through the grid inmany different ways and that it is not immediately obvious that we will always endup with the same map A n,m → Y . Still, this is the case, because if both f i,j : A i,j → Y and g i,j : A i,j → Y are obtained by repeatedly taking our favourite pushforwards forthese little squares, in some order, then one easily proves that f i,j = g i,j by inductionon n = i + j .The second (and final) worry is that the grid decomposing a small mould squareinto one-step mould squares is not uniquely determined. Clearly, we have no choicein how to travel in the horizontal direction, but in the vertical direction we havesome choice, coming from the following fact. If S ⊆ T ⊆ ∆ n is an inclusion ofcofibrant sieves and we write it as a sequence S = S ⊆ S ⊆ S ⊆ . . . ⊆ S n = T of cofibrant sieves where each S i +1 is obtained from S i by adding a single m -simplexwhose faces belonged to S i , then this sequence is far from unique. However, any twosuch sequences can be obtained from each other by repeatedly applying permutationsof the following form: if we have such a sequence and somewhere in this sequencewe have U ⊆ V ⊆ W where V is obtained from U by adding some k -simplex and W is obtained from W by adding some l -simplex and the k -simplex is not a faceof the l -simplex (so that the boundaries of both the l -simplex and k -simplex werealready present in U ), then we can replace this by U ⊆ V (cid:48) ⊆ W where V (cid:48) is obtainedfrom U by adding the l -simplex and W is obtained from V (cid:48) by adding the k -simplex.Since our answer to the first worry tells us that we may always assume that the wayone finds the lift against a grid as above is by computing all the lifts A i,j → Y inlexicographic order, we end up with the following statement that we need to prove: FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 143
Lemma 12.4.
Suppose
U, V, V (cid:48) , W are cofibrant sieves as above and we have a liftingproblem of the form: ( U, θ ) ( U, (cid:104) i, ±(cid:105) ∗ θ ) Y ( W, θ ) ( W, (cid:104) i, ±(cid:105) ∗ θ ) X p then the solutions obtained by decomposing the left hand square as in the diagrambelow on the left or as in the one below on the right coincide. ( U, θ ) ( U, (cid:104) i, ±(cid:105) ∗ θ ) Y ( V, θ ) ( V, (cid:104) i, ±(cid:105) ∗ θ )( W, θ ) ( W, (cid:104) i, ±(cid:105) ∗ θ ) X p ( U, θ ) ( U, (cid:104) i, ±(cid:105) ∗ θ ) Y ( V (cid:48) , θ ) ( V (cid:48) , (cid:104) i, ±(cid:105) ∗ θ )( W, θ ) ( W, (cid:104) i, ±(cid:105) ∗ θ ) X p Proof.
Without loss of generality we may assume that θ is the empty traversal, thereason being that the solutions for general θ are obtained by pushing forward thesolutions for θ = (cid:104)(cid:105) .Let us write α : [ k ] → [ n ] for the k -simplex in V but not in U and β : [ l ] → [ n ] forthe l -simplex in W but not in V . Then the solutions for the lifting problem in the left
44 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS hand squares in both diagrams above are determined by the following mould cubes:( U, (cid:104)(cid:105) ) ( U, (cid:104) i, ±(cid:105) )( ∂ ∆ k , (cid:104)(cid:105) ) ( ∂ ∆ k , (cid:104) i, ±(cid:105) · α )( V, (cid:104)(cid:105) ) ( V, (cid:104) i, ±(cid:105) )(∆ k , (cid:104)(cid:105) ) (∆ k , (cid:104) i, ±(cid:105) · α ) ( α, (cid:104)(cid:105) ) ( α, (cid:104)(cid:105) )( α, (cid:104)(cid:105) ) ( α, (cid:104)(cid:105) ) ( U, (cid:104)(cid:105) ) ( U, (cid:104) i, ±(cid:105) )( ∂ ∆ l , (cid:104)(cid:105) ) ( ∂ ∆ l , (cid:104) i, ±(cid:105) · β )( V (cid:48) , (cid:104)(cid:105) ) ( V (cid:48) , (cid:104) i, ±(cid:105) )(∆ l , (cid:104)(cid:105) ) (∆ l , (cid:104) i, ±(cid:105) · β ) ( β, (cid:104)(cid:105) ) ( β, (cid:104)(cid:105) )( β, (cid:104)(cid:105) ) ( β, (cid:104)(cid:105) ) ( V, (cid:104)(cid:105) ) ( V, (cid:104) i, ±(cid:105) )( ∂ ∆ l , (cid:104)(cid:105) ) ( ∂ ∆ l , (cid:104) i, ±(cid:105) · β )( W, (cid:104)(cid:105) ) ( W, (cid:104) i, ±(cid:105) )(∆ l , (cid:104)(cid:105) ) (∆ l , (cid:104) i, ±(cid:105) · β ) ( β, (cid:104)(cid:105) ) ( β, (cid:104)(cid:105) )( β, (cid:104)(cid:105) ) ( β, (cid:104)(cid:105) ) ( V (cid:48) , (cid:104)(cid:105) ) ( V (cid:48) , (cid:104) i, ±(cid:105) )( ∂ ∆ k , (cid:104)(cid:105) ) ( ∂ ∆ k , (cid:104) i, ±(cid:105) · α )( W, (cid:104)(cid:105) ) ( W, (cid:104) i, ±(cid:105) )(∆ k , (cid:104)(cid:105) ) (∆ k , (cid:104) i, ±(cid:105) · α ) ( α, (cid:104)(cid:105) ) ( α, (cid:104)(cid:105) )( α, (cid:104)(cid:105) ) ( α, (cid:104)(cid:105) ) But since the second and third as well as the first and fourth have the same frontface, the solutions will coincide. (What this says is that because the k -simplex and FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 145 l -simplex have at most parts of their boundary in common, the the lifting problemsare independent from each other and can be solved in either order.) (cid:3) To summarise the discussion so far, any map p which has the right lifting prop-erty with respect to horn squares, has unambiguous lifts against general small mouldsquares. Note that these lifts will automatically satisfy the horizontal and verticalconditions for having the right lifting property against the triple category M , becausethe lifts do not depend on the way we divide a small mould square into a grid ofone-step mould squares or on the way we traverse that grid. That means that any re-quirements on the lifts against the horn squares needed for them to extend to a uniqueeffective Kan fibration structure should come from the perpendicular condition. Sowe will now have a look at this condition. Remark 12.5.
From now on, we will often think of the perpendicular condition onthe lifts against the small mould squares in M as expressing as a stability conditionwith respect to base change or pullback. Indeed, we will often refer to it as a basechange condition . The reason is that in M both the morphisms of HDRs in the yz -plane as well as in the xz -plane are cartesian. This means that in a small mould cubelike D CD (cid:48) C (cid:48) B AB (cid:48) A (cid:48) β we can think of the front face of the cube as the result of pulling back the face at theback along β : B (cid:48) → B . Indeed, from now on we will often draw such a situation asfollows D (cid:48) C (cid:48) D CB (cid:48) A (cid:48) B A. β The reader is supposed to keep in mind that there is a cube connecting the twosquares, but we will only draw a dotted arrow to prevent our diagrams from becomingtoo cluttered.Let us call a small mould square stable if it its induced lift is compatible with theinduced lift of any base change of that same square. In fact, it will be convenient tohave a relativised notion of stability. So assume S is a class of morphisms in ∆ such
46 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS that if [ n (cid:48) ] β (cid:48) (cid:15) (cid:15) α (cid:48) (cid:47) (cid:47) [ m (cid:48) ] β (cid:15) (cid:15) [ n ] α (cid:47) (cid:47) [ m ]is a pullback diagram in ∆ with β monic, then α ∈ S implies α (cid:48) ∈ S (think S = { face maps } ∪ { identities } or S = { degeneracy maps } ∪ { identities } ). Then asmall mould square D CB A will be called S -stable if for any commutative diagram of the form D C YD (cid:48) C (cid:48) B A XB (cid:48) A (cid:48) p in which the cube is a “small” base change cube along ( α, τ ): B (cid:48) → B with α ∈ S , theinduced map A (cid:48) → Y can be obtained by composing the induced map A → Y with A (cid:48) → A . The next step is to find necessary and sufficient conditions on the fillers forthe horn squares to ensure that any small mould square is S -stable.First of all, it is clearly necessary and sufficient if every one-step mould square is S -stable. Indeed, if in a grid any little square is S -stable, then so is the entire square.But note that the pullback of a one-step mould square need no longer be a one-stepmould square: both in the horizontal and the vertical direction the number of stepsmay increase. (By the way, it may also become 0 in one of the two directions, inwhich case the stability condition is vacuously satisfied. So without loss of generalitywe may always assume this does not happen.) FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 147
Lemma 12.6.
Suppose ψ = τ ∗ θ and we have situation as follows: ( S, θ ) (
S, ψ ) Y ( S, (cid:104)(cid:105) ) ( S, τ )( T, θ ) (
T, ψ ) X ( T, (cid:104)(cid:105) ) ( T, τ ) , p where the cube is a base change cube along (1 , τ ): ( T, (cid:104)(cid:105) ) → ( T, θ ) . Then the inducedlifts ( T, τ ) → Y and ( T, ψ ) → Y are compatible. This means that, since the bottomof the cube is a pushout, the induced lifts determine each other by composition andpushout, respectively.Proof. We prove the statement of the lemma by induction on the length of the tra-versal τ . Note that the case τ = (cid:104)(cid:105) is vacuously true.In case where τ has length 1, we can regard both the front and the back of thecube as a vertical composition of one-step mould squares. In that case the statementfollows from the definition of the induced lifts for one-step mould squares.Now write τ = σ ∗ ρ where σ has length 1 and consider the following situation:( S, (cid:104)(cid:105) ) ( S, σ )( T, (cid:104)(cid:105) ) ( T, σ )( S, (cid:104)(cid:105) ) ( S, ρ ) (
S, τ )( T, (cid:104)(cid:105) ) ( T, ρ ) (
T, τ )( S, θ ) (
S, ρ ∗ θ ) ( S, ψ ) Y ( T, θ ) (
T, ρ ∗ θ ) ( T, ψ ) X (1 ,ρ )(1 ,θ ) (1 ,θ ) p We should imagine that we are given a map (
T, θ ) → Y and we want to push itforward to a map ( T, ψ ) → Y . Now, by induction hypothesis, the fact that thestatement holds in case τ has length 1, and the earlier lemmas about grids, we cancompute this as follows: take the induced lift ( T, ρ ) → Y , push that down to map( T, ρ ∗ θ ) → Y , restrict that to a map ( T, (cid:104)(cid:105) ) → Y , take the induced lift ( T, σ ) → Y
48 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS and then push that all the way down to a map (
T, ψ ) → Y . But the latter can bedone in two steps: push the map ( T, σ ) → Y down to ( T, τ ) → Y and then push itfurther down to ( T, ψ ) → Y . This means the end result ( T, ψ ) → Y coincides withtaking the induced lift ( T, τ ) → Y and pushing that down. (cid:3) Lemma 12.7.
If all one-step mould squares starting from the empty traversal are S -stable, then so are all one-step mould squares.Proof. Imagine that we have a one-step mould square(
S, θ ) ( S, (cid:104) i, ±(cid:105) ∗ θ )( T, θ ) ( T, (cid:104) i, ±(cid:105) ∗ θ )and we pull it back along ( α, σ ) with α ∈ S . Writing θ · α = ψ ∗ σ , we get four smallmould squares:( α ∗ S, (cid:104)(cid:105) ) ( α ∗ S, (cid:104) i, ±(cid:105) · α ) ( S, (cid:104)(cid:105) ) ( S, (cid:104) i, ±(cid:105) )( α ∗ T, (cid:104)(cid:105) ) ( α ∗ T, (cid:104) i, ±(cid:105) · α ) ( T, (cid:104)(cid:105) ) ( T, (cid:104) i, ±(cid:105) )( α ∗ S, ψ ) ( α ∗ S, (cid:104) i, ±(cid:105) · α ∗ ψ ) ( S, θ ) ( S, (cid:104) i, ±(cid:105) ∗ θ )( α ∗ T, ψ ) ( α ∗ T, (cid:104) i, ±(cid:105) · α ∗ ψ ) ( T, θ ) ( T, (cid:104) i, ±(cid:105) ∗ θ ) (1 ,ψ ) ( α,
1) (1 ,θ )( α,σ ) Recall from Remark 12.5 that the dotted arrows indicate that the squares are con-nected by small mould cubes and note that the small mould cubes determined bythe dotted arrows going down have pushouts at their bottom faces. We are givena map (
T, θ ) → Y and asked to compare the induced maps ( T, (cid:104) i, ±(cid:105) ∗ θ ) → Y and( α ∗ T, (cid:104) i, ±(cid:105) · α ∗ ψ ) → Y . The previous lemma tells us that both induced maps canbe computed by taking the induced maps ( α ∗ T, (cid:104) i, ±(cid:105) · α ) → Y and ( T, (cid:104) i, ±(cid:105) ) → Y and then pushing these down. So if the square on the top right is S -stable, then sois the square on the bottom right. (cid:3) Lemma 12.8.
Suppose S ⊆ T ⊆ ∆ m are cofibrant sieves, α : ∆ n → ∆ m is monic and R S ∆ nα T FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 149 is bicartesian. Suppose moreover that we have situation as follows: ( S, (cid:104)(cid:105) ) ( S, θ ) Y ( R, (cid:104)(cid:105) ) ( R, θ · α )( T, (cid:104)(cid:105) ) ( T, θ ) X (∆ n , (cid:104)(cid:105) ) (∆ n , θ · α ) , p where the cube is a base change cube along ( α, (cid:104)(cid:105) ): (∆ n , (cid:104)(cid:105) ) → ( T, (cid:104)(cid:105) ) . Then the inducedlifts ( T, θ ) → Y and (∆ n , θ · α ) → Y are compatible. This means that, since the leftand right hand faces of the cube are pushouts, the induced lifts determine each otherby composition and pushout, respectively.Proof. We prove this by induction on the number k of simplices in T but not in S (which coincides with the number of simplices in ∆ n but not in R ). Note that thecase k = 0 is trivial.In case k = 1, we have R = ∂ ∆ n . We prove the desired statement by inductionon the length of θ . Note that because α is monic, θ · α cannot have greater lengththan θ . The case θ = (cid:104)(cid:105) is again trivial, while the case where θ has length 1 followsimmediately from the way the lifts for horn squares induce lifts for one-step mouldsquare starting from the empty traversal. Now write θ = τ ∗ σ where τ has length 1and consider: ( R, (cid:104)(cid:105) ) ( R, τ · α )(∆ n , (cid:104)(cid:105) ) (∆ n , τ · α )( R, (cid:104)(cid:105) ) ( R, σ · α ) ( R, θ · α )(∆ n , (cid:104)(cid:105) ) (∆ n , σ · α ) (∆ n , θ · α )( S, (cid:104)(cid:105) ) ( S, σ ) (
S, θ ) Y ( T, (cid:104)(cid:105) ) ( T, σ ) (
T, θ ) X (1 ,σ · α )( α, (cid:104)(cid:105) ) p We should imagine that we are given a map ( T, (cid:104)(cid:105) ) → Y and we want to push itforward to a map ( T, θ ) → Y . Now, by induction hypothesis, the fact that the
50 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS statement holds in case θ has length 1 (and 0), and the earlier lemmas about grids,we can compute this by taking the induced lift (∆ n , σ · α ) → Y , pushing it down to amap ( T, σ ) → Y , then taking the induced lift (∆ n , τ · α ) → Y and then pushing thatdown to a map ( T, θ ) → Y . But the latter can be done in two steps: pushing it downto (∆ n , θ · α ) → Y and then pushing it further down to ( T, θ ) → Y . This means itcoincides with taking the induced lift (∆ n , θ · α ) → Y and pushing that down.Having proved the statement for k = 1, we now do the induction step. So write S ⊆ S (cid:48) ⊆ T where S (cid:48) is obtained from S by adding one simplex, so that we have apicture as follows: ∂ ∆ n (cid:48) R S ∆ n (cid:48) R (cid:48) S (cid:48) ∆ n T α (cid:48) α in which all squares are bicartesian. This gives us the following situation:( ∂ ∆ n (cid:48) , (cid:104)(cid:105) ) ( ∂ ∆ n (cid:48) , θ · αα (cid:48) )(∆ n (cid:48) , (cid:104)(cid:105) ) (∆ n (cid:48) , θ · αα (cid:48) )( R, (cid:104)(cid:105) ) ( R, θ · α )( R (cid:48) , (cid:104)(cid:105) ) ( R (cid:48) , θ · α )(∆ n , (cid:104)(cid:105) ) (∆ n , θ · α )( S, (cid:104)(cid:105) ) ( S, θ ) Y ( S (cid:48) , (cid:104)(cid:105) ) ( S (cid:48) , θ )( T, (cid:104)(cid:105) ) ( T, θ ) X ( α (cid:48) , (cid:104)(cid:105) )( α, (cid:104)(cid:105) ) p So if we have a map ( T, (cid:104)(cid:105) ) → Y and we wish to push it forward to ( T, θ ) → Y , thenwe can do this in two steps: first we can compute ( S (cid:48) , θ ) → Y and then compute( T, θ ) → Y . The former we can compute by finding the lift (∆ n (cid:48) , θ · αα (cid:48) ) → Y and thenpushing it down, which also can be done in two steps, yielding a map ( R (cid:48) , θ · α ) → Y and then a map ( S (cid:48) , θ ) → Y . Given this map ( S (cid:48) , θ ) → Y , the induction hypothesis FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 151 tells us that we can compute the desired map (
T, θ ) → Y from the map ( R (cid:48) , θ · α ) → Y we computed along the way by taking its induced lift (∆ n , θ · α ) → Y and then pushingthat down. In other words, we take the induced map (∆ n , θ · α ) → Y and then pushthat down, thus showing the induction step. (cid:3) Lemma 12.9.
If all horn squares are S -stable, then so are all one-step mould squaresstarting from the empty traversal.Proof. Suppose we have a one-step mould square starting from the empty traversal,like ( S, (cid:104)(cid:105) ) ( S, (cid:104) i, ±(cid:105) )( T, (cid:104)(cid:105) ) ( T, (cid:104) i, ±(cid:105) )which we want to pull it back along some map, say ( α, (cid:104)(cid:105) ) with α ∈ S (note that thesecond component has to be the empty traversal). Let β : [ m ] → [ n ] be the m -simplexwhich we need to add to S to obtain T , and consider the pullback:∆ m (cid:48) ∆ m ∆ n (cid:48) ∆ nα (cid:48) β (cid:48) γ βα (note that β is monic, so this pullback exists provided the images of α and β havesome overlap: which we may assume without loss of generality, because otherwise α ∗ S = α ∗ T and the stability condition is trivially satisfied). Note that α (cid:48) ∈ S .We again get four mould squares, where the dotted arrows again indicate somebase changes:( γ ∗ S, (cid:104)(cid:105) ) ( γ ∗ S, (cid:104) i, ±(cid:105) · γ ) ( ∂ ∆ m , (cid:104)(cid:105) ) ( ∂ ∆ m , (cid:104) i, ±(cid:105) · β )(∆ m (cid:48) , (cid:104)(cid:105) ) (∆ m (cid:48) , (cid:104) i, ±(cid:105) · γ ) (∆ m , (cid:104)(cid:105) ) (∆ m , (cid:104) i, ±(cid:105) · β )( α ∗ S, (cid:104)(cid:105) ) ( α ∗ S, (cid:104) i, ±(cid:105) · α ) ( S, (cid:104)(cid:105) ) ( S, (cid:104) i, ±(cid:105) )( α ∗ T, (cid:104)(cid:105) ) ( α ∗ T, (cid:104) i, ±(cid:105) · α ) ( T, (cid:104)(cid:105) ) ( T, (cid:104) i, ±(cid:105) ) ( β (cid:48) , (cid:104)(cid:105) ) ( α (cid:48) , (cid:104)(cid:105) ) ( β, (cid:104)(cid:105) )( α, (cid:104)(cid:105) ) Note that the base change cubes for the arrows going down have pushouts as their leftand right faces. We are given a map ( T, (cid:104)(cid:105) ) → Y and asked to compare the inducedmaps ( T, (cid:104) i, ±(cid:105) ) → Y and ( α ∗ T, (cid:104) i, ±(cid:105)· α ) → Y . The previous lemma tells us that bothinduced maps can be computed by first taking the induced maps (∆ m (cid:48) , (cid:104) i, ±(cid:105)· γ ) → Y
52 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS and (∆ m , (cid:104) i, ±(cid:105) · β ) → Y and then pushing them down. So if the square on the topright is S -stable, then so is the square on the bottom right. (cid:3) From Lemma 12.7 and Lemma 12.9 we deduce:
Proposition 12.10.
If we equip a map p : Y → X with lifts against horn squareswhich are S -stable, then all the induced lifts against small mould squares will be S -stable. What does this mean for effective Kan fibrations? To equip a map p : Y → X withthe structure of an effective Kan fibration, it will be (necessary and) sufficient to findlifts against horn squares so that the induced lifts against small mould squares are S -stable for both S = { face maps } ∪ { identities } and S = { degeneracy maps } ∪{ identities } . So the proposition tells us that we need to find lifts against horn squareswhich are stable relative to both classes. But lifts against horn squares are alwaysstable relative to the first class (faces plus identities), because d ∗ i ∂ ∆ n is always themaximal sieve. So we have: Theorem 12.11.
The following notions of fibred structure are isomorphic: • Being an effective Kan fibration. • To assign to each map all systems of lifts against horn squares which arestable along degeneracy maps.
Let us now try to unwind what that means concretely: lifts for horn squares whichare stable along degeneracies map. First of all, for each n there are 2( n + 1) hornsquares as follows: ( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, ±(cid:105) )(∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, ±(cid:105) ) , and these can be pulled back along s j : ∆ n +1 → ∆ n . The case where j = i is specialand we will postpone discussion of that case.In case j (cid:54) = i we have the following cartesian morphism of HDRs:∆ n +1 ∆ n +2 ∆ n +1 ∆ n ∆ n +1 ∆ ns j d i ∗ /d i ∗ +1 s j ∗ s i ∗ s j d i /d i +1 s i where i ∗ = i + 1 if j < i and i ∗ = i if j > i , while j ∗ = j if j < i and j ∗ = j + 1if j > i . This means that if we pull back the horn square above along s j , then we FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 153 obtain a vertical composition of three one-step mould squares, as follows:( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, ±(cid:105) ) ( S n +1 j , (cid:104)(cid:105) ) ( S n +1 j , (cid:104) i ∗ , ±(cid:105) )(∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, ±(cid:105) ) (Λ n +1 j , (cid:104)(cid:105) ) (Λ n +1 j , (cid:104) i ∗ , ±(cid:105) )( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, ±(cid:105) ) ( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i ∗ , ±(cid:105) )(∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, ±(cid:105) ) (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i ∗ , ±(cid:105) )( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, ±(cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, ±(cid:105) ) X. d j +1 d j s j p Here we have used the abbreviation S n +1 j : = s ∗ j ∂ ∆ n = Λ n +1 j,j +1 ∪ ( d j ∩ d j +1 ) (that is, itis ∆ n +1 with the interior as well as the j th and ( j +1)st faces missing). Note that ourrecipe for computing the lifts against the first two squares in the vertical compositionon the top right in the diagram tells us to solve the original lifting problem (because s j .d j = s j .d j +1 = 1). In other words, we can phrase the compatibility condition forthe case j (cid:54) = i as follows: if f : (∆ n , (cid:104) i, ±(cid:105) ) → Y is our chosen solution to the liftingproblem ( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, ±(cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, ±(cid:105) ) X, y pg x then our chosen solution (∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) → Y to the lifting problem( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) Y (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) X y (cid:48) pg.s j x.s j ∗ should be f.s j ∗ , where y (cid:48) is the map which is y · s j ∗ on S n +1 j and f on both d j and d j +1 .The case j = i is special, because then the pullback of the horn square grows ina horizontal direction as well. In this case it will be convenient to treat the positiveand negative case separately. So let do the positive case first. As we have seen in the
54 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS proof of Theorem 10.9, we have the following cartesian morphism of HDRs:∆ n +1 ∆ n +2 (cid:104) ( i + 1 , +) , ( i, +) (cid:105) ∆ n +1 ∆ n ∆ n +1 ∆ n . s i d i ι [ s i ,s i +1 ] [ s i +1 ,s i ] s i d i s i This means that we have a picture as follows in which we have pulled back the hornsquare below along s i and decomposed the result into a grid of six one-step mouldsquares. ( S n +1 i , (cid:104)(cid:105) ) ( S n +1 i , (cid:104) i, + (cid:105) ) ( S n +1 i , (cid:104) ( i + 1 , +) , ( i, +) (cid:105) )(Λ n +1 i , (cid:104)(cid:105) ) (Λ n +1 i , (cid:104) i, + (cid:105) ) (Λ n +1 i , (cid:104) ( i + 1 , +) , ( i, +) (cid:105) )( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i, + (cid:105) ) ( ∂ ∆ n +1 , (cid:104) ( i + 1 , +) , ( i, +) (cid:105) )(∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i, + (cid:105) ) (∆ n +1 , (cid:104) ( i + 1 , +) , ( i, +) (cid:105) )( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, + (cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, + (cid:105) ) X. s i y pg x Let us first consider the left column in the grid above. Note that the if we pull backthe first square along d i +1 , we get the original square back, while if we pull back thesecond square along d i , we get a map which trivialises in the horizontal direction.For that reason the left column gives us the following compatibility condition: if f : (∆ n , (cid:104) i, + (cid:105) ) → Y is our chosen solution to the lifting problem( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, + (cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, + (cid:105) ) X, y pg x then our chosen solution (∆ n +1 , (cid:104) i, + (cid:105) ) → Y to the lifting problem( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i, + (cid:105) ) Y (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i, + (cid:105) ) X y (cid:48) pg.s i x.s i +1 should be f.s i +1 , where y (cid:48) is the map which is y · s i +1 on S n +1 i and f on d i +1 . FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 155
We now turn to the column on the right. Note that if we pull back the first squarealong d i +1 , the square trivialises in the horizontal direction, while if we pull back thesecond square along d i , we get the original horn square back. Therefore the righthand column gives us the following compatibility condition: if f : (∆ n , (cid:104) i, + (cid:105) ) → Y isour chosen solution to the lifting problem( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, + (cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, + (cid:105) ) X, y pg x then our chosen solution to the lifting problem( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i + 1 , + (cid:105) ) Y (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i + 1 , + (cid:105) ) X y (cid:48) pg.s i x.s i should be f.s i , where y (cid:48) is the map which is y · s i on S n +1 i and f on d i .Now let us do the negative case. In that case we have following cartesian morphismof HDRs: ∆ n +1 ∆ n +2 (cid:104) ( i, − ) , ( i + 1 , − ) (cid:105) ∆ n +1 ∆ n ∆ n +1 ∆ n . s i d i +2 ι [ s i +1 ,s i ] [ s i ,s i +1 ] s i d i +1 s i The situation we now have to look at is the one where we pull the horn square at thebottom of diagram below back along s i and decompose the result into six one-stepmould squares.( S n +1 i , (cid:104)(cid:105) ) ( S n +1 i , (cid:104) i + 1 , −(cid:105) ) ( S n +1 i , (cid:104) ( i, − ) , ( i + 1 , − ) (cid:105) )(Λ n +1 i , (cid:104)(cid:105) ) (Λ n +1 i , (cid:104) i + 1 , −(cid:105) ) (Λ n +1 i , (cid:104) ( i, − ) , ( i + 1 , − ) (cid:105) )( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i + 1 , −(cid:105) ) ( ∂ ∆ n +1 , (cid:104) ( i, − ) , ( i + 1 , − ) (cid:105) )(∆[ n + 1] , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i + 1 , −(cid:105) ) (∆ n +1 , (cid:104) ( i, − ) , ( i + 1 , − ) (cid:105) )( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, −(cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, −(cid:105) ) X. s i p
56 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
Both columns in the grid will again determine a compatibility condition and to seewhat they are, we start of by considering the left hand column. Note that the pullbackof the first square along d i +1 is trivial in the horizontal direction, while if we pull backthe second square along d i , we get our original horn square back. So the compatibilitycondition becomes this: if f : (∆ n , (cid:104) i, −(cid:105) ) → Y is our chosen solution to the liftingproblem ( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, −(cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, −(cid:105) ) X, y pg x then our chosen solution to the lifting problem( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i + 1 , −(cid:105) ) Y (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i + 1 , −(cid:105) ) X y (cid:48) pg.s i x.s i should be f.s i , where y (cid:48) is the map which is y · s i on S n +1 i and f on d i .Finally, if we consider the right hand column, then the pullback of the first squarealong d i +1 gives us the original horn square back, while the pullback of the secondsquare along d i trivialises in the horizontal diagram. Therefore this column yields thefollowing compatibility condition: if f is our chosen solution to the lifting problem( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, −(cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, −(cid:105) ) X, y pg x then our chosen solution to the lifting problem( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i, −(cid:105) ) Y (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i, −(cid:105) ) X y (cid:48) pg.s i x.s i +1 should be f.s i +1 , where y (cid:48) is the map which is y · s i +1 on S n +1 i and f on d i +1 .To summarise the entire discussion, let us write for each n ∈ N : A n = (cid:8) ( i, j, i + 1 , j ) : i, j ≤ n, j < i (cid:9) ∪ (cid:8) ( i, j, i, j + 1) : i, j ≤ n, j > i (cid:9) ∪ (cid:8) ( i, i, i, i + 1) : i ≤ n (cid:9) ∪ (cid:8) ( i, i, i + 1 , i ) : i ≤ n (cid:9) Theorem 12.12.
The following notions of fibred structure are isomorphic: • To be an effective Kan fibration.
FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 157 • To assign to a map p : Y → X lifts against horn squares, in such a way thatfor any n ∈ N , ( i, j, i ∗ , j ∗ ) ∈ A n and ± ∈ { + , −} : if f is our chosen solutionto the lifting problem ( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, ±(cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, ±(cid:105) ) X, y pg x then our chosen solution to the lifting problem ( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) Y (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) X y (cid:48) pg.s j x.s j ∗ should be f.s j ∗ , where y (cid:48) is the map which is y · s j ∗ on S n +1 j and f on thefaces d k with k ∈ { j, j + 1 } − { i ∗ } . Remark 12.13.
The second bullet in the theorem above is really a lifting conditionagainst horn squares, not horns inclusions. We have seen at the beginning of thesection that every horn inclusion is induced by some horn square, but for inner hornsthis horn square is not unique (it is for outer horns). Indeed, if Λ ni is an inner horn,then it is induced by two horn squares, one coming from ( i, +) and one coming from( i − , − ), and our notion of an effective Kan fibration may choose different lifts forthese two horn squares.A result similar to Theorem 12.12 holds for effective right (left) fibrations: thisnotion of fibred structure is equivalent to having the right lifting property with re-spect to horn squares with positive (negative) orientation, with the lifts satisfyingthe compatibility in the second item of that theorem. In fact, since there are nocompatibility conditions relating the horn squares with different polarity, we obtain: Proposition 12.14.
In the category of notions of fibred structure being an effectiveKan fibration is the categorical product of being a effective left fibration and being aneffective right fibration.
Local character and classical correctness.
From the characterisation ofeffective Kan fibrations in Theorem 12.12 we can deduce that our notion of being aneffective Kan fibration is both local and classically correct.
Corollary 12.15.
The notion of being an effective Kan fibration is a local notion offibred structure.Proof.
Suppose p : Y → X is a map and for every pullback of p along a map x : ∆ n → X we have a stable choice of a structure as in Theorem 12.12. Then, if we are given a
58 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS lifting problem as in ( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, ±(cid:105) ) Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, ±(cid:105) ) X, y pg x then we may pull p back along x and we get:( ∂ ∆ n , (cid:104)(cid:105) ) ( ∂ ∆ n , (cid:104) i, ±(cid:105) ) Y x Y (∆ n , (cid:104)(cid:105) ) (∆ n , (cid:104) i, ±(cid:105) ) ∆ n +1 X. y x ∗ p pg x So, using the lifting structure of x ∗ p , we obtain a map (∆ n , (cid:104) i, ±(cid:105) ) → Y x which wemay compose with Y x → Y . In this way we obtain a lift against p . We still have tocheck that such lifts satisfy the condition in Theorem 12.12.So imagine that we wish to solve( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) Y (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) X y (cid:48) pg.s j x.s j ∗ with ( i, j, i ∗ , j ∗ ) ∈ A n and where y (cid:48) is the map which is y · s j ∗ on S n +1 j and f on thefaces d k with k ∈ { j, j + 1 } − { i ∗ } . The recipe we were given is that we write this as( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) Y x.s j ∗ Y (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) ∆ n +2 X, y (cid:48) ( x.s j ∗ ) ∗ p pg.s j x.s j ∗ find the induced lift (∆ n +1 , (cid:104) i ∗ , + (cid:105) ) → Y x.s j ∗ and compose with Y x.s j ∗ → Y . But wemay write the pullback in the previous diagram as the composition of two pullbacks,as follows:( ∂ ∆ n +1 , (cid:104)(cid:105) ) ( ∂ ∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) Y x.s j ∗ Y x Y (∆ n +1 , (cid:104)(cid:105) ) (∆ n +1 , (cid:104) i ∗ , ±(cid:105) ) ∆ n +2 ∆ n +1 X, y (cid:48) ( x.s j ∗ ) ∗ p x ∗ p pg.s j s j ∗ x By our stability assumption, this means that the composition of the induced with(∆ n +1 , (cid:104) i ∗ , + (cid:105) ) → Y x.s j ∗ with Y x.s j ∗ → Y x is the induced lift (∆ n +1 , (cid:104) i ∗ , + (cid:105) ) → Y x against x ∗ p . But the latter is f.s j ∗ , because x ∗ p has lifts satisfying the condition inTheorem 12.12. This means that p has lifts satisfying that condition as well, finishingthe proof. (cid:3) FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 159
Corollary 12.16.
In a classical metatheory, every map which has the right liftingproperty against horns (a Kan fibration in the usual sense) can be equipped with thestructure of an effective Kan fibration.Proof.
Suppose that we have a map which has the right lifting property against allhorns. Because a lifting problem against a horn has at most one degenerate solution(see Proposition B.1), we may always choose the degenerate solution if it exists.In that case our lifts will satisfy the condition in Theorem 12.12, because it saysthat under certain circumstances we should choose a degenerate solution. But byalways choosing the unique degenerate solution (if it exists), this will automaticallybe satisfied. (cid:3)
Remark 12.17.
We again have similar results for effective left and right fibrations.Indeed, proofs which are almost identical to the ones of Corollary 12.15 and Corollary12.16 yield: • Being an effective right (left) fibration is a local notion of fibred structure. • In a classical metatheory, a map can be equipped with the structure of aneffective right (left) fibration if and only if it has the right lifting propertyagainst horn inclusions Λ ni → ∆ n with i (cid:54) = 0 (with i (cid:54) = n ), that is, if and onlyif it is a right (left) fibration in the usual sense.
60 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
Conclusion
To summarise, we have introduced the effective Kan fibrations, a new structuredanalogue of the Kan fibrations in simplicial sets, and we have proved the followingresults:(1) Effective Kan fibrations are closed under push forward along effective Kanfibrations.(2) Being an effective Kan fibration is a local notion of fibred structure.(3) In a classical metatheory, a map can be equipped with the structure of aneffective Kan fibration if and only if it has the right lifting property againsthorn inclusions.Despite the length of this document, this is only the beginning of a new effectiveapproach towards the theory of simplicial sets. In particular, we have left open thecorrect constructive answers to the following questions:(1) Are effective Kan fibrations the right class in an AWFS?(2) Is every uniform Kan fibration an effective Kan fibration?(3) Is the notion of a uniform Kan fibration local?(4) Are effective Kan fibrations closed under retracts?Note that in a classical metatheory the answer to all these questions is yes , so thatwithin a constructive metatheory the answer can never be no (it could be that aconstructive metatheory leaves some of these questions open).The reason a classical metatheory answers questions (2) – (4) in the affirmative isbecause in a classical metatheory the maps which can be equipped with a uniformor effective Kan fibration structure are precisely the Kan fibrations. As for question(1), we have an argument showing that the effective Kan fibrations are the right classin an AWFS, but at present it relies on an ineffective version of the small objectargument. However, we fully expect that this can be made constructive, so that alsothe constructive answer to the first question is yes .Questions (2) and (3) are the most pressing questions for understanding the preciserelationship of our work to that of Gambino and Sattler. We have seen that everyeffective Kan fibration is also a uniform Kan fibration in their sense, and it seemsto us quite unlikely that, within a constructive metatheory, this implication can bereversed, but we have no proof of this. One way in which this could be proved is byshowing that that the statement that uniform Kan fibrations are local is constructivelyunprovable. If, however, this can be shown constructively, this would allow one torevive the approach by Gambino and Sattler.Another way of showing that the uniform and effective Kan fibrations are, con-structively, distinct structures is by proving that there can be no constructive proofof the statement that the effective Kan fibrations are closed under retracts (this beingtrue for the uniform Kan fibrations). Right now we have no constructive reason to FFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS 161 believe that effective Kan fibrations are closed under retracts, and we expect that nosuch thing exists.Apart from these specific questions, the larger issues are whether there can be aconstructive proof of the existence of an algebraic model structure and a model ofhomotopy type theory (including univalence and higher-inductive types) based on thenotion of an effective Kan fibration. We have some positive results in that direction,and we hope to be able to report on this very soon in a document of smaller size.
62 EFFECTIVE KAN FIBRATIONS IN SIMPLICIAL SETS
Appendices A. Axioms
In this appendix we will collect the axioms for a Moore category and a dominancethat play a role in this paper. The reader can think of these as our version of theOrton-Pitts axioms [OP18] (see also [GS17]; [FB19]).A.1.
Moore structure.
Our first ingredient is a suitable notion of Moore paths.
Definition A.1.
Let E be a category with finite limits. A Moore structure on E consists of the following data:(1) We have a pullback-preserving endofunctor M on E together with naturaltransformations r : 1 E → M , s, t : M → E , and µ : M t × s M → M turningevery object X in E in the object of objects of an internal category, with M X as the object of arrows. Note the order in which µ takes its arguments:it is not in the way categorical composition is usually written. The reasonis that we think of µ as concatenation of paths rather than as categoricalcomposition and we write it as such.(2) There is a natural transformation Γ: M → M M making (
M, s,
Γ) into acomonad.(3) There is a strength α X,Y : X × M Y → M ( X × Y ), that is, α is a naturaltransformation making( X × Y ) × M Z M (( X × Y ) × Z ) X × ( Y × M Z ) X × M ( Y × Z ) M ( X × ( Y × Z )) ∼ = α X × Y,Z ∼ =1 X × α Y,Z α X,Y × Z X × M Y M ( X × Y ) M Y α X,Y p Mp commute. In addition, all the previous structure is strong, so the followingdiagrams commute as well: X × M Y M ( X × Y ) X × Y X × Y α × s × t s t × r r X × M Y t × s M Y M ( X × Y ) t × s M ( X × Y ) X × M Y M ( X × Y ) × µ ( α. ( p ,p ) ,α. ( p ,p )) µα X × M Y M ( X × Y ) X × M M Y M ( X × M Y ) M M ( X × Y ) . α × Γ Γ α Mα (4) We have the following axioms for the connection Γ (interaction with r, t ):Γ .r = rM.r, tM. Γ = r.t, M t.
Γ = θ X .α X, . ( t, M !) , with θ X being the iso M p : M ( X × → M X .(5) And, finally, we have the following distributive law (interaction between Γand µ ): Γ .µ = µ. ( M µ.ν. (Γ .p , θ MX .α MX, . ( p , M ! .p )) , Γ .p ): M X × X M X → M M X with ν being the natural transformation (in this case M M X × MX M M X → M ( M X × X M X )) induced by preservation of pullbacks. This condition canbe visualized as follows. When p, q ∈ M X are composable Moore paths as inthe left-hand size of the diagram, then Γ .µ ( p, q ) is defined by Γ( p ) and Γ( q )in the following way:(36) zyx y zpq Γ .µ ( p, q ) = zyx y zzpq Γ( q )Γ( p ) α ( q, M ! p )Whenver a category E is equipped with structure thus described, we call E a categorywith Moore structure , or a Moore category for short.
Remark A.2.
The notion of a path object category from [BG12] can be obtainedfrom this by dropping the coassociativity axiom for Γ as well as the distributive law,whilst adding a symmetry τ (see below). Remark A.3.
As observed in [BG12], the fact that M preserves pullbacks meansthat the entire strength is determined by the maps α X : = X × M M ( X × M X. α X, θ X The reason for this is that the outer rectangle and right hand square in X × M Y M ( X × Y ) M YX × M M X M α X,Y X × M ! p Mp Mp M ! α X p M ! are pullbacks. And, if we wish, axioms (4-6) can also be formulated as follows: thereis a natural transformation α X : X × M → M X with M ! .α X = p : X × M → M
1, and, in addition, the following diagrams commute: X × M M XX × X α × s × t s t ∼ = p × r r X × M × M M X × X M XX × M M X × µ ( α. ( p ,p ) ,α. ( p ,p )) µα X × M M XX × M M M X × M M M M ( X × M M M X. α × Γ Γ( α. ( p ,M ! .p ) ,p ) ∼ = Mα Finally, we have the following axioms for the interaction between the connection Γand the category structure:Γ .r = rM.r,tM a. Γ = r.t,M t.
Γ = α. ( t, M !) , Γ .µ = µ. ( M µ.ν X . (Γ .p , α MX . ( p , M ! .p )) , Γ .p ) . Definition A.4.
We will call a Moore structure two-sided if it also comes equippedwith a map Γ ∗ : M → M M turning (
M, t, Γ ∗ ) into a strong comonad, and such thatthe following equations hold:(37) Γ ∗ .r = rM.r,s. Γ ∗ = r.s,M s. Γ ∗ = α. ( s, M !) , Γ ∗ .µ = µ. (Γ ∗ .p , M µ.ν. ( α M . ( p , M ! .p ) , Γ ∗ .p )) . This has the effect that if we switch s and t and define µ ∗ : = µ. ( p , p ), then we get asecond Moore structure. We will also require that µ is both left and right cancellative,and that we have the sandwich equation: M µ.ν. (Γ ∗ , Γ) = α. (1 , M !): M → M M (which also implies
M µ ∗ .ν. (Γ , Γ ∗ ) = α. (1 , M !)). Definition A.5.
A two-sided Moore structure will be called symmetric if it alsocomes equipped with a natural transformation τ : M → M such that τ.τ = 1 ,τ.r = r,s.τ = t,t.τ = s, Γ ∗ = τ M.M τ. Γ .τ, while also the following diagrams commute: X × M M XX × M M X α × τ τα M X t × s M X M XM X t × s M X M X µ ( τ.p ,τ.p ) τµ Remark A.6.
There is a bit of redundancy in the previous definition, in that Γ ∗ = τ M.M τ. Γ .τ implies the equations (37) above. Example A.7.
The following examples from [BG12] all satisfy the axioms for asymmetric Moore category.(1) The category of topological spaces with
M X = (cid:88) r ∈ R ≥ X [0 ,r ] , the space of Moore paths.(2) The category of small groupoids with M X = X I , where I is the interval groupoid containing two objects and one arrow x → y for any pair of objects ( x, y ). In fact, this also defines a symmetric Moorestructure on the category of small categories.(3) The category of chain complexes over a ring R .For more details we refer to [BG12, Section 5].A.2. Dominance.
The second ingredient is a dominance [Ros86].
Definition A.8. A dominance on a category E is a class of monomorphism Σ in E satisfying the following three properties:(1) every isomorphism is in Σ and Σ is closed under composition.(2) every pullback of a map in Σ again belongs to Σ.(3) the category Σ cart of maps in Σ and pullback squares between them has aterminal object 1 → Σ.For some of our arguments it will be convenient to assume the following two addi-tional axioms:(1) The elements in Σ are closed under finite unions; that is, 0 → X alwaysbelongs to Σ and whenever A → X and B → X belong to Σ, then so does A ∪ B → X .(2) The morphism r X : X → M X belongs to Σ for any object X . B. Degenerate horn fillers are unique
The purpose of this appendix is to show that horn filling problems have at mostone degenerate filler, in the following sense:
Proposition B.1.
If both x · σ and x · σ are fillers for Λ ni X ∆ n where σ : ∆ n → ∆ k and σ : ∆ n → ∆ l are epimorphisms in ∆ different from theidentity, then x · σ = x · σ . The proof strategy that we will follow here was suggested to us by Christian Sattler.The (constructive) argument relies on the following lemma (see [BM15, Lemma 5.6]):
Lemma B.2.
Suppose that we have a diagram of the form • • •• • • f g f exhibiting f as a retract of g , while g has a section. Then the right hand square is anabsolute pushout.Proof. (Of Proposition B.1.) It suffices to consider the case where σ = s i and σ = s j with i < j .Because i (cid:54) = j + 1 in at least one of the following diagrams the dotted arrow exists:Λ nk ∆ n − ∆ n ∆ n − ∆ n − ∆ n − ∆ n − . s j − d i s j s i s j − d i s i Λ nk ∆ n − ∆ n ∆ n − ∆ n − ” ∆ n − ∆ n − s i d j +1 s i s j s i d j s j − In either case, the previous lemma implies that both the inner and outer square inΛ nk ∆ n ∆ n − ∆ n − ∆ n − s i s j s i s j − are pushouts, from which the proposition follows. (cid:3) ibliography [AAG03] Michael Gordon Abbott, Thorsten Altenkirch, and Neil Ghani. “Cate-gories of Containers”. In: Foundations of Software Science and Compu-tational Structures, 6th International Conference, FOSSACS 2003 Heldas Part of the Joint European Conference on Theory and Practice ofSoftware, ETAPS 2003, Warsaw, Poland, April 7-11, 2003, Proceedings .2003, pp. 23–38. doi : .[AU17] Danel Ahman and Tarmo Uustalu. “Taking Updates Seriously”. In: Pro-ceedings of the 6th International Workshop on Bidirectional Transfor-mations co-located with The European Joint Conferences on Theory andPractice of Software, BX@ETAPS 2017, Uppsala, Sweden, April 29, 2017 .Ed. by Romina Eramo and Michael Johnson. Vol. 1827. CEUR WorkshopProceedings. CEUR-WS.org, 2017, pp. 59–73.[AW09] Steve Awodey and Michael A. Warren. “Homotopy theoretic models ofidentity types”. In:
Math. Proc. Cambridge Philos. Soc. doi : .[BCH19] Marc Bezem, Thierry Coquand, and Simon Huber. “The Univalence Ax-iom in Cubical Sets”. In: J. Autom. Reasoning doi : .[BCP15] Marc Bezem, Thierry Coquand, and Erik Parmann. “Non-constructivityin Kan simplicial sets”. In: . Vol. 38. LIPIcs. Leibniz Int. Proc. Inform. SchlossDagstuhl. Leibniz-Zent. Inform., Wadern, 2015, pp. 92–106.[Bec69] Jon Beck. “Distributive laws”. In: Sem. on Triples and Categorical Ho-mology Theory (ETH, Z¨urich, 1966/67) . Springer, Berlin, 1969, pp. 119–140.[BG11] Benno van den Berg and Richard Garner. “Types are weak ω -groupoids”.In: Proc. Lond. Math. Soc. (3) doi : .[BG12] Benno van den Berg and Richard Garner. “Topological and simplicialmodels of identity types”. In: ACM Trans. Comput. Log. doi : .[BG16] John Bourke and Richard Garner. “Algebraic weak factorisation systemsI: Accessible AWFS”. In: J. Pure Appl. Algebra doi : . [BM15] Benno van den Berg and Ieke Moerdijk. “W-types in homotopy type the-ory”. In: Math. Structures Comput. Sci. doi : .[BR70] Jean B´enabou and Jacques Roubaud. “Monades et descente”. In: C. R.Acad. Sci. Paris S´er. A-B
270 (1970), A96–A98.[CHS19] Thierry Coquand, Simon Huber, and Christian Sattler. “Homotopy Canon-icity for Cubical Type Theory”. In: . Ed. by Herman Geuvers. Vol. 131. LIPIcs.Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2019, 11:1–11:23. doi : .[Cis14] Denis-Charles Cisinski. “Univalent universes for elegant models of homo-topy types”. arXiv:1406.0058. 2014.[Coh+17] Cyril Cohen et al. “Cubical Type Theory: A Constructive Interpretationof the Univalence Axiom”. In: FLAP doi : .[FB19] Daniil Frumin and Benno van den Berg. “A homotopy-theoretic model offunction extensionality in the effective topos”. In: Math. Struct. Comput.Sci. doi : .[Gar09] Richard Garner. “Understanding the small object argument”. In: Appl.Categ. Structures doi : .[GH19] Nicola Gambino and Simon Henry. “Towards a constructive simplicialmodel of univalent foundations”. arXiv:1905.06281. 2019.[GK13] Nicola Gambino and Joachim Kock. “Polynomial functors and polynomialmonads”. In: Math. Proc. Cambridge Philos. Soc. doi : .[GS17] Nicola Gambino and Christian Sattler. “The Frobenius condition, rightproperness, and uniform fibrations”. In: J. Pure Appl. Algebra
Calculus of fractions and homotopy theory .Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer-Verlag New York, Inc., New York, 1967, pp. x+168.[Hen19] Simon Henry. “A constructive account of the Kan-Quillen model structureand of Kan’s Ex ∞ Twenty-five years of constructive type theory (Venice,1995) . Vol. 36. Oxford Logic Guides. Oxford Univ. Press, New York, 1998,pp. 83–111.
70 BIBLIOGRAPHY [Hub19] Simon Huber. “Canonicity for Cubical Type Theory”. In:
J. Autom. Rea-soning doi : .[KL18] C. Kapulkin and P.L Lumsdaine. “The simplicial model of univalent foun-dations (after Voevodsky)”. arXiv:1211.2851. Accepted for publication inthe Journal of the European Mathematical Society . 2018.[Lum10] Peter Lefanu Lumsdaine. “Weak ω -categories from intensional type the-ory”. In: Log. Methods Comput. Sci. doi : .[Lur09] Jacob Lurie. Higher topos theory . Vol. 170. Annals of Mathematics Stud-ies. Princeton University Press, Princeton, NJ, 2009, pp. xviii+925. doi : .[May67] J. Peter May. Simplicial objects in algebraic topology . Van Nostrand Math-ematical Studies, No. 11. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967, pp. vi+161.[MP00] Ieke Moerdijk and Erik Palmgren. “Wellfounded trees in categories”. In:
Proceedings of the Workshop on Proof Theory and Complexity, PTAC’98(Aarhus) . Vol. 104. 1-3. 2000, pp. 189–218. doi : .[Nor19] Paige North. Type-theoretic weak factorization systems . 2019. arXiv: .[OP18] Ian Orton and Andrew M. Pitts. “Axioms for Modelling Cubical TypeTheory in a Topos”. In:
Logical Methods in Computer Science doi : .[Pro13] The Univalent Foundations Program. Homotopy type theory—univalentfoundations of mathematics . The Univalent Foundations Program, Prince-ton, NJ; Institute for Advanced Study (IAS), Princeton, NJ, 2013, pp. xiv+589.[PW02] John Power and Hiroshi Watanabe. “Combining a monad and a comonad”.In: vol. 280. 1-2. Coalgebraic methods in computer science (Amsterdam,1999). 2002, pp. 137–162. doi : .[Rie11] Emily Riehl. “Algebraic model structures”. In: New York J. Math. ftp://ftp.disi.unige.it/pub/person/RosoliniG/papers/coneit.ps.gz . PhDthesis. Carnegie Mellon University, 1986.[Sat18] Christian Sattler. “The equivalence extension property and model struc-tures”. arXiv:1704.06911. 2018.[Shu19] Michael Shulman. “All ( ∞ , doi : https://doi.org/10.17863/CAM.16245https://doi.org/10.17863/CAM.16245