The effective model structure and \infty-groupoid objects
Nicola Gambino, Simon Henry, Christian Sattler, Karol Szumi?o
aa r X i v : . [ m a t h . C T ] F e b The effective model structure and ∞ -groupoid objects Nicola Gambino Simon Henry Christian Sattler Karol Szumi loFebruary 12, 2021
Abstract
For a category E with finite limits and well-behaved countable coproducts, we constructa model structure, called the effective model structure, on the category of simplicial objectsin E , generalising the Kan–Quillen model structure on simplicial sets. We then prove that theeffective model structure is left and right proper and satisfies descent in the sense of Rezk. Asa consequence, we obtain that the associated ∞ -category has finite limits, colimits satisfyingdescent, and is locally Cartesian closed when E is, but is not a higher topos in general. We alsocharacterise the ∞ -category presented by the effective model structure, showing that it is thefull sub-category of presheaves on E spanned by Kan complexes in E , a result that suggests aclose analogy with the theory of exact completions. Introduction
Context and motivation.
Over the past two decades, there has been an explosion of interest inthe connections between model categories and higher categories [Cis20, GK17, JT07, Lur09, Rez01,Szu17]. This line of research led to the reformulation of significant parts of modern homotopy theoryin terms of higher category theory, the development of higher topos theory [TV05, Lur09] and is ofgreat importance for Homotopy Type Theory and the Univalent Foundations programme [AW09,BM18b, GK17, KL12, Shu19]. Central to these developments are model structures on categories ofsimplicial objects, i.e., functor categories of the form s E = [∆ op , E ], where E is a category [GJ99]. Inparticular, the category of simplicial sets equipped with the Kan–Quillen model structure [Qui67]can be understood as a presentation of the ∞ -category of spaces, while categories of simplicialpresheaves and sheaves (i.e., simplicial objects in a Grothendieck topos) equipped with the Rezkmodel structure [Rez10] and the Joyal–Jardine model structure [Bro73, Joy84, Jar96] can be seenas presentations of ∞ -toposes and their hypercompletions, respectively [DHI04, Lur09].The main contribution of this paper is to construct a new model structure, which we callthe effective model structure , on categories of simplicial objects s E , assuming that E is merely acountably lextensive category, i.e., a category with finite limits and countable coproducts, wherethe latter are required to be van Kampen colimits [CLW93, Rez10]. The effective model structure isdefined so that when E = Set , we recover the Kan–Quillen model structure on simplicial sets [Qui67].We also prove several results on the effective model structure and its associated ∞ -category, whichwe discuss below.The initial motivation for this work was the desire to establish whether our earlier work on theconstructive Kan–Quillen model structure [Hen19, GS17, GSS19, Sat17] could be developed further
1o as to obtain a new model structure on categories of simplicial sheaves. Indeed, in [Hen19, GSS19]we worked with simplicial sets without using the law of excluded middle and the axiom of choice,thus opening the possibility of replacing them with simplicial objects in a Grothendieck topos. Aswe explored this idea, we realised that the resulting argument admitted not only a clean presentationin terms of enriched weak factorisation systems [Rie14, Chapter 13], but also a vast generalisation.In fact, the existence of the effective model structure may be a surprise to some readers, sinceassuming E to be countably lextensive is significantly weaker than assuming it to be a Grothendiecktopos and covers many more examples (such as the category of countable sets and the category ofschemes). In particular, our arguments do not require the existence of all small colimits, (local)Cartesian closure and local presentability, which are ubiquitous in the known constructions of modelstructures.One reason for the interest in the effective model structure is that, when E is a Grothendiecktopos, the effective model structure on s E differs from the known model structures on simplicialsheaves and provides the first example of a peculiar combination of higher categorical structure.Indeed, the associated ∞ -category has finite limits, colimits that satisfy descent and is locally Carte-sian closed, but is neither a higher Grothendieck topos [Lur09] nor a higher elementary topos in thesense of [Shu17, Ras18], since its 0-truncation does not always have a subobject classifier (see Ex-ample 11.8). In this case, the effective model structure satisfies most of the axioms for a modeltopos [Rez10], but is not combinatorial. One key point here is that the effective model structure isnot cofibrantly generated in the usual sense, but only in an enriched sense.This situation can be understood by analogy with the theory of exact completions in ordinarycategory theory [CV98]. There, it is known that the exact completion of a (Grothendieck) toposneed not be a (Grothendieck) topos [Men03]. Indeed, we believe that the effective model structurewill provide a starting point for the development of a homotopical counterpart of the theory ofexact completions. As a first step in this direction, we prove that the ∞ -category associated tothe effective model structure on s E is the full subcategory of the ∞ -category of presheaves on E spanned by Kan complexes in E , mirroring a corresponding description of the exact completionof E in [HT96]. We also make a conjecture (Conjecture 13.2) on the relation between the effectivemodel structure and ∞ -categorical exact completions, which we leave for future work. In the longterm, we hope that our work could be useful for the definition of a higher categorical version of theeffective topos [Hyl82], which can be described as an exact completion [Car95].Finally, our results may be of interest also in Homotopy Type Theory, since they help to clarifyhow the simplicial model of Univalent Foundations [KL12], in which types are interpreted as Kancomplexes, is related to the so-called setoid model of type theory [Hof97], in which types areinterpreted as types equipped with an equivalence relation, which are obtained by versions of theexact completion [EP17]. Indeed, we expect that the effective model structure may lead to newmodels of Homotopy Type Theory, another topic that we leave for future research. Main results.
In order to outline our main results, let us briefly describe the effective modelstructure, whose fibrant objects are to be thought of as Kan complexes, or ∞ -groupoids, in E . Inorder to describe the fibrations of the effective model structure, recall that, for E ∈ E , we have afunctor Hom sSet ( E, − ) : s E → sSet ( ∗ )sending X ∈ s E to the simplicial set defined by Hom sSet ( E, X ) n = Hom( E, X n ), for [ n ] ∈ ∆. Wecan then define a map in s E to be a fibration in s E if its image under the functor in ( ∗ ) is a Kanfibration in sSet for every E ∈ E . Trivial fibrations are defined analogously. Our main results arethe following: 2 Theorem 9.9, asserting the existence of the effective model structure, whose fibrations andtrivial fibrations are defined as above; • Proposition 10.4 and Corollary 12.18, asserting that the effective model structure is right andleft proper, respectively, and Proposition 10.1, showing that homotopy colimits in s E satisfydescent; • Theorem 10.3 asserting that the ∞ -category Ho ∞ ( s E ) associated to the effective model struc-ture has finite limits and α -small colimits satisfying descent when E is α -lextensive, andTheorem 10.5, showing that Ho ∞ ( s E ) is also locally Cartesian closed when E is so. • Theorem 13.1, characterising the ∞ -category associated to the effective model structure.Along the way, we prove several other results of independent interest. For example, we charac-terise completely the cofibrations of the effective model structure, which do not coincide with allmonomorphisms (Theorem 4.6) and we compare the effective model structure with model structuresstudied in relation to Elmendorf’s Theorem (Theorem 11.7). Novel aspects.
This paper differs significantly from our work in [Hen19, GSS19, Sat17] in bothscope and technical aspects. Regarding scope, apart from generalising the existence of the modelstructure from the case E = Set to that of a general countably lextensive category E , here we discussa number of topics that are not even mentioned for the case E = Set in our earlier work, such asthe structure and characterisation of the ∞ -category associated to the effective model structure,the discussion of descent and the connections with Elmendorf’s theorem.Regarding the technical aspects, even if the general strategy for proving the existence of theeffective model structure is inspired from the case E = Set in [GSS19], several new ideas are necessaryto implement it to the general case, as we explain below. This strategy involves in three steps.First, we introduce the notions of a (trivial) fibration in s E as above and establish the existence ofa fibration category structure on the category of Kan complexes (assuming in fact only that E hasfinite limits). Secondly, we construct the two weak factorisation systems of the model structure,one given by cofibrations and trivial fibrations and one given by trivial cofibrations and fibrations.Thirdly, we show that weak equivalences (as determined by the two weak factorisation systems)satisfy 2-out-of-3 by proving the so-called Equivalence Extension Property (Proposition 8.3).In order to realise this plan, we prove several results that are not necessary for E = Set .We mention only the key ones. First, we develop a new version of the enriched small objectargument (Theorem 3.14), which does not require existence of all colimits. In order to achieve it, weanalyse the colimits required for our applications and prove that they exist in a countably lextensivecategory, exploiting crucially that some of the maps involved are complemented monomorphisms.Secondly, we show that the fibration category structure, where fibrations are defined as above,agrees with the weak factorisation systems, defined in terms defined in terms of enriched liftingproperties (Proposition 4.1). Thirdly, we obtain a characterisation of cofibrations in categories ofsimplicial objects (Theorem 4.6), which requires a new, purely categorical, argument that is entirelydifferent to the one used in [Hen19, GSS19, Sat17]. Finally, new ideas are required in the proof ofthe Equivalence Extension Property (Proposition 8.3). For this, we need to construct explicitlydependent products (i.e., pushforward) functors along cofibrations (Theorem 6.5), which are notguaranteed to exist since E is not assumed to be locally Cartesian closed. The existence of thesepushforward functors may be considered as a pleasant surprise since they are essential for ourargument and no exponentials are assumed to be present in E .3he existence of the effective model structure is independent from that of the constructive Kan–Quillen model structure on simplicial sets [GSS19, Hen19]. Actually, the use of ideas of enrichedcategory theory here, especially for expressing stronger versions of the lifting properties usuallyphrased in terms of mere existence of diagonal fillers, makes explicit some of the informal conventionsadopted in [GSS19,Hen19] when treating the case E = Set . Also, the proofs in [GSS19,Hen19] makeuse of structure on
Set that is not available in a countably lextensive category and therefore cannotbe interpreted as taking place in the so-called internal logic of E [Joh02, Section D1.3]. Even when E is a Grothendieck topos, carrying out the proofs in the internal language of E [MLM92, Chapter 6]would not make explicit the structure under consideration, thus making it more difficult for theresults to be accessible and applicable. Outline of the paper.
The paper is organised in four parts. The first, including only Section 1,establishes the fibration category structure. The second, including Sections 2, 3 and 4, introducesthe two weak factorisation systems, having first developed an appropriate version of the smallobject argument. The third, including Sections 5, 6, 7, 8 and 9, establishes the existence of theeffective model structure, by constructing pushforward functors and establishing the Frobenius andEquivalence Extension Property. The fourth, including Sections 10, 11, 12 and 13, proves the keyproperties of the effective model structure, namely descent and properness, their ∞ -categoricalcounterparts, and characterises its associated ∞ -category. Throughout the paper, we omit theproofs that can be carried out with minor modifications from [Hen19, GSS19], but include the onesthat require new ideas. Remark.
The material in this paper is developed within ZFC set theory. Some of the material,however, can be developed also in a constructive setting (see footnotes and Appendix A for details).
Acknowledgements.
We are grateful to Andr´e Joyal for questions and discussions, which inparticular led to Theorem 13.1. Nicola Gambino and Karol Szumi lo gratefully acknowledge thesupport of the US Air Force Office for Scientific Research under agreement FA8655-13-1-3038. SimonHenry was partially supported by an NSERC Discovery Grant. Christian Sattler was supported bySwedish Research Council grant 2019-03765.
This section develops some simplicial homotopy theory in a category E with finite limits. Thecategory of simplicial objects in E is defined by letting s E = def [∆ op , E ] . In Definition 1.3 we introduce the notion of a fibration in s E with which we shall work throughoutthe paper. This notion is defined using the enrichment of s E in sSet and generalises that of aKan fibration in sSet . The main result of this section, Theorem 1.7, establishes a structure of afibration category on the category of fibrant objects in s E . For applications throughout the paper,we also establish a fiberwise version of this fibration category in Theorem 1.9. We also introducethe notion of a pointwise weak equivalence (Definition 1.6), which provides the weak equivalencesof these fibration categories. In the subsequent sections we will extend these results to obtain theeffective model structure on s E , under the stronger assumption that E is countably lextensive. Theweak equivalences of the effective model structure will not be the pointwise weak equivalences ingeneral, although the two notions will coincide for maps between fibrant objects.4et us recall how the category s E is enriched over sSet with respect to the Cartesian monoidalstructure. For a finite simplicial set K and X ∈ s E , we define K ⋔ X ∈ s E via the end formula( K ⋔ X ) m = def R [ n ] ∈ ∆ X ( K × ∆[ m ]) n n . (1.1)For X, Y ∈ s E , the simplicial hom-object is then defined by letting Hom sSet ( X, Y ) m = def Hom
Set ( X, ∆[ m ] ⋔ Y ). (1.2)This makes s E into a sSet -enriched category so that the formula in (1.1) gives the cotensor (overfinite simplicial sets) with respect to the enrichment. Without further assumptions on E , s E doesnot admit all cotensors or tensors over simplicial sets. We often identify an object E ∈ E with theconstant simplicial object with value E . For example, for E ∈ E and Y ∈ s E we write Hom sSet ( E, Y ).Note that Hom sSet ( E, Y ) m = Hom Set ( E, Y m ),Hom sSet ( E, K ⋔ Y ) = K ⋔ Hom sSet ( E, Y ).The sSet -enrichment allows us to define a notion of a homotopy between morphisms of s E . Givenmaps f , f : X → Y in s E (or one of its slice categories), a homotopy H from f to f , written H : f ∼ f , is a map H : X → ∆[1] ⋔ Y (1.3)that restricts to f on { } → ∆[1] and to f on { } → ∆[1]. It is constant if it factors throughthe canonical map ∆[0] ⋔ Y → ∆[1] ⋔ Y , in which case f = f . Note that we can regard H as amap ∆[1] → Hom sSet ( X, Y ). This generalises the usual notion of homotopy in simplicial sets. Foreach E ∈ E , the functor Hom sSet ( E, − ) preserves homotopies because it preserves the cotensor with∆[1].We need some definitions to introduce the notions of a Kan fibration and trivial Kan fibration in s E . For a finite simplicial set K , we define the evaluation functor ev K : s E → E via the end formulaev K ( X ) = X ( K ) = def R [ n ] ∈ ∆ X K n n . (1.4)We will usually write X ( K ) rather than ev K ( X ) for brevity. However, in some situations thenotation ev K ( X ) will be more convenient, see the definition of pullback evaluation below. The endabove exists since, by the finiteness of K , it can be constructed from finite limits. For example, X (∆[ n ]) = X n and X (Λ k [2]) = X × X X . Also note that X ( K ) = ( K ⋔ X ) and X ( K × ∆[ m ]) =( K ⋔ X ) m . Remark 1.1.
There are two alternative ways of viewing the evaluation functor. First, since E hasfinite limits, we can consider X ( K ) as the value on K of the right Kan extension of X : ∆ op → E along the inclusion of ∆ into the category of finite simplicial sets. Secondly, seeing E as a Set -enriched category, we can view X ( K ) as a weighted limit, namely the limit of X , viewed as a diagramin E , weighted by K , viewed as a diagram in Set . Both of these observations show that X ( K ) iscontravariantly functorial in K . Here and in the following we use subscripts to indicate to which category the hom-objects under considerationbelong.
5e write b ev for the pullback evaluation functor, which is the result of applying the so-calledLeibniz construction [RV14] to the two-variable functor ev, i.e., the functor sending a map i : A → B between finite simplicial sets and a morphism f : X → Y of s E to b ev i ( f ) : ev A ( X ) → ev B ( X ) × ev B ( Y ) ev A ( Y ) in E (1.5)also written as b ev i ( f ) : X ( A ) → X ( B ) × Y ( B ) Y ( A ). Remark 1.2.
We adopt the convention of prefixing with ‘pullback’ (or ‘pushout’) the name of atwo-variable functor to indicate the result of applying the Leibniz construction to it. So for example,we shall say pushout product for what is also referred to as Leibniz product or corner product.We use standard notation for the sets of boundary inclusions and horn inclusions, I sSet = { ∂ ∆[ n ] → ∆[ n ] | n ≥ } and J sSet = { Λ k [ n ] → ∆[ n ] | n ≥ k ≥ , n > } . (1.6) Definition 1.3.
We say that a morphism in s E is • a trivial Kan fibration if its pullback evaluations with all maps in I sSet are split epimorphisms; • a Kan fibration if its pullback evaluations with all maps in J sSet are split epimorphisms.Explicitly, a map f : X → Y in s E is a Kan fibration if the morphism X (∆[ n ]) → X (Λ k [ n ]) × Y (Λ k [ n ]) Y (∆[ n ])in E has a section, for all n ≥ k ≥ n >
0. For Y = 1, this means that the morphism X (∆[ n ]) → X (Λ k [ n ])has a section, for all n ≥ k ≥ n >
0, in which case we say that X is a Kan complex . Notethat for E = Set , these definitions reduce to the standard notions of a Kan fibration, trivial Kanfibration and a Kan complex in simplicial sets. In the following, we shall frequently write fibration , trivial fibration and fibrant object , as we do not consider other notions of fibrations.Although we have not yet introduced cofibrations and trivial cofibrations in s E , we can use thestandard classes of cofibrations and trivial cofibrations in sSet , which are the saturations of thegenerating sets I sSet and J sSet , respectively.The next proposition characterises fibrations and trivial fibrations by reducing them to thecorresponding notions in sSet in terms of the sSet -enrichment of s E , defined in (1.2). Proposition 1.4.
Let f : X → Y be a map in s E . Then f is a (trivial) fibration if and only if, forall E ∈ E , the map Hom sSet ( E, f ) : Hom sSet ( E, X ) → Hom sSet ( E, Y ) is a (trivial) fibration in sSet .Proof. Note that the functors X ( − ) : sSet op → E and Hom sSet ( − , X ) : E op → sSet are contravari-antly adjoint. Thus for all maps i : A → B between finite simplicial sets there is a bijectivecorrespondence between the lifting problems A Hom sSet ( E, X ) X ( B ) B Hom sSet ( E, Y ) E X ( A ) × Y ( A ) Y ( B ) b ev i ( f ) E = X ( A ) × Y ( A ) Y ( B )).If i : A → B is a map of finite simplicial sets and p : X → Y is a morphism of s E , then we definethe pullback cotensor of i and p (cf. Remark 1.2) as the induced morphism i b ⋔ p : B ⋔ X → ( A ⋔ X ) × A ⋔ Y ( B ⋔ X ). Lemma 1.5. (i)
The pullback cotensor in s E of a cofibration between finite simplicial sets and a fibration is afibration. If the given cofibration or fibration is trivial, then the result is a trivial fibration. (ii) Fibrations and trivial fibrations in s E are closed under composition, pullback, and retract. (iii) Let f : X → Y and g : Y → Z be morphisms of s E . If f : X → Y and gf : X → Z are trivialfibrations, then so is g : Y → Z .Proof. All the statements are proved in the same way: they hold for simplicial sets (see, e.g., [Qui67,Theorem II.3.3]) and transfer to s E using Proposition 1.4. Note that transferring (i) from sSet to s E relies on the fact that Hom sSet ( E, − ) preserves pullbacks and cotensors and hence pullbackcotensors. Definition 1.6.
Let f : X → Y in s E . We say that f is a pointwise weak equivalence ifHom sSet ( E, f ) : Hom sSet ( E, X ) → Hom sSet ( E, Y )is a weak equivalence in sSet for all E ∈ E .For the next theorem, we use the definition of a fibration category as stated in [GSS19, Sec-tion 1.6]. Theorem 1.7.
Let E be category with finite limits. Then pointwise weak equivalences, Kan fibra-tions and trivial Kan fibrations equip the category of Kan complexes in s E with the structure of afibration category.Proof. Trivial fibrations are exactly the fibrations that are weak equivalences because this holds in sSet . We need to verify the following axioms.(F1) s E has a terminal object and all objects are fibrant, which follows directly from the definitions.(F2) Pullbacks along fibrations exist because E (and hence s E ) has all finite limits. Moreover,fibrations and acyclic fibrations are closed under pullback by point (ii) of Lemma 1.5.(F3) Every morphism factors as a weak equivalence followed by a fibration. By [Bro73, p. 421,Factorization lemma] it suffices to construct a path object, i.e., a factorisation of the diagonal X → X × X . Such factorisation is given by the cotensor X → ∆[1] ⋔ X → X × X . ApplyingHom sSet ( E, − ) to this factorisation givesHom sSet ( E, X ) → ∆[1] ⋔ Hom sSet ( E, X ) → Hom sSet ( E, X ) × Hom sSet ( E, X ) Constructively, part (i) is true in sSet by [GSS19, Corollary 1.3.4], part (ii) is evident and part (iii) is [GSS19,Lemma 1.3.6]. sSet ( E, X ) into a weak equivalencefollowed by a fibration in sSet (since Hom sSet ( E, X ) is a Kan complex by Proposition 1.4).See, e.g., [GJ99, p. 43]. Hence X → ∆[1] ⋔ X → X × X is also such factorisation in s E .(F4) Weak equivalences satisfy 2-out-of-6, which follows since this property holds in sSet .In view of our development in Section 8, we generalise Theorem 1.7 to the case of a slice of s E over a simplicial object X , which we write s E ↓ X . We then define s E ↡ X to be the full subcategoryof s E ↓ X spanned by the fibrations over X .First of all, let us recall that the enrichment of s E in simplicial sets, including the cotensorwith finite simplicial sets, descends to its slices. For ( A, f ) , ( B, g ) ∈ s E ↓ X , the hom-objectHom sSet (( A, f ) , ( B, g )) is the pullback of Hom sSet ( A, B ) along the map f : 1 → Hom sSet ( A, X ).The cotensor of (
A, f ) ∈ s E ↓ X by a finite simplicial set K is the pullback of K ⋔ A along the map X → K ⋔ X (using the fact that the monoidal unit in sSet is the terminal object). As before, foreach E , the functor Hom sSet ( E, − ) : s E ↓ X → sSet ↓ Hom sSet ( E, X ) preserves these cotensors.
Lemma 1.8.
Let X ∈ s E . The pullback cotensor properties in part (i) of Lemma 1.5 hold in s E ↓ X as well.Proof. This follows from their validity in s E , i.e., part (i) of Lemma 1.5 and the stability of fibrationsand trivial fibration under pullback, i.e., part (ii) of Lemma 1.5. Theorem 1.9.
Let X ∈ s E . Then pointwise weak equivalences, fibrations and trivial fibrationsequip the category s E ↡ X with the structure of a fibration category.Proof. All axioms are verified by the same argument as in the proof of Theorem 1.7. For (F3),we use Lemma 1.8 which is a fiberwise version of part (i) of Lemma 1.5 used in the proof ofTheorem 1.7.We conclude this section with a basic observation on homotopy equivalences.
Proposition 1.10.
Homotopy equivalences in s E (and in particular, in s E ↓ X for all X ∈ s E ) arepointwise weak equivalences.Proof. The functors Hom sSet ( E, − ) preserve homotopies and hence also homotopy equivalences.Thus the conclusion follows from the fact that homotopy equivalences are weak equivalences in sSet . This section, Section 3 and Section 4 constitute the second part of the paper, whose ultimategoal is to construct two weak factorisation systems on s E , whose right classes of maps are thefibrations and trivial fibrations of Section 1, assuming that s E is a countably lextensive category.This section recalls some basic facts about lextensive categories. Throughout it, we consider afixed category with finite limits E and study diagrams in E over an indexing category D . Whenconvenient, we will regard cones under such diagrams as diagrams over the category D ⊲ , obtainedby adding a new terminal object ⋆ to D . We start by recalling the general notion of van Kampencolimit [Lur09, Rez10] in our setting. 8 efinition 2.1. Let Y • : D → E be a diagram and assume Y ⋆ = colim d ∈ D Y d is its colimit in E .We say that Y ⋆ is(i) universal , if it is preserved by pullbacks, i.e., if for every map X ⋆ → Y ⋆ , X ⋆ is the colimit ofthe induced diagram X d = X ⋆ × Y ⋆ Y d .(ii) effective , if given a Cartesian natural transformation X → Y , the diagram X has a colimit X ⋆ , and all the squares X d X ⋆ Y d Y ⋆ are pullback squares, i.e., the extended natural transformation over D ⊲ is also Cartesian.(iii) van Kampen , if it is both universal and effective. Lemma 2.2.
A colimit Y ⋆ = colim d ∈ D Y d in E is van Kampen if and only if it is preserved by thepseudo-functor E op → Cat sending each X ∈ E to the slice category E ↓ X (with morphisms actingby pullbacks). In other words, the slice category E ↓ Y ⋆ is the pseudo-limit lim d ∈ D ( E ↓ Y d ) .Proof. Pullback along the structure morphisms of Y ⋆ induces a functor E ↓ Y ⋆ → lim d ( E ↓ Y d ). Weneed to show that this functor is an equivalence if and only if the colimit of Y • is a van Kampencolimit.An object of lim d ( E ↓ Y d ) can be identified with a Cartesian transformation X → Y . If colimitsof diagrams Cartesian over Y • exist then taking the colimit yields a left adjoint to the functorabove. Conversely, we claim that if this pullback functor has a left adjoint, then the left adjointcomputes the colimits of diagrams that are Cartesian over Y • . Indeed, assume that the pullbackfunctor E ↓ Y ⋆ → lim d ( E ↓ Y d ) has a left adjoint X • X ⋆ , and let Z be an arbitrary object of E . Amap X ⋆ → Z in E is the same as a map X ⋆ → Z × Y ⋆ in E ↓ Y ⋆ , which by the adjunction formulais the same as a natural transformation X d → Z × Y d over Y • , but this is exactly the same as anatural transformation X d → Z in E , and hence this shows that X ⋆ is the colimit of X d .Now, Y ⋆ is universal if and only if the counit of this adjunction is an isomorphism and it iseffective if and only if the unit is an isomorphism. Hence this shows that the colimit Y ⋆ of Y • is vanKampen if and only if the pullback functor described above has a left adjoint such that the unitand counit of the adjunction are isomorphisms, i.e., if and only if it is an equivalence.For example, an initial object 0 is always vacuously effective and it is universal if and only if itis strict , i.e., if there is a morphism X →
0, then X is initial itself. Instead, a coproduct Y ⋆ = ` d Y d is van Kampen if and only if it is universal and disjoint , i.e., Y d × Y ⋆ Y d ′ is initial for d = d ′ . Thiscan be seen inspecting the proof of [CLW93, Proposition 2.14].For S ∈ Set and X ∈ E , we write S · X for the tensor of X with S , when it exists. If E hascountable coproducts, then this tensor exists for countable S and can be defined as S · X = a s ∈ S X . (2.1)9he global sections functor E (1 , − ) : E →
Set has a partial left adjoint, defined by mapping acountable set S to S = def S · a s ∈ S . (2.2)We extend this notation to diagram categories in a levelwise fashion: if E has countable coproductsand D a small category, then the levelwise global sections functor E D → Set D has a partial leftadjoint, sending a levelwise countable diagram K ∈ Set D to K ∈ E D , which is defined by levelwiseapplication of S S . These functors will be used frequently in the paper. For example, we willuse them in Section 4 to transfer the sets of boundary inclusions and horn inclusions in (1.6) from sSet to s E , so as to obtain generating sets for weak factorisation systems in s E . We establish someof their basic properties in the next lemmas. Lemma 2.3. If E is countably lextensive, then for every countable set S and X ∈ E , we have S × X ∼ = S · X , naturally in S .Proof. Since E is countably lextensive, it is countably distributive. Thus, product with X preservescountable coproducts, in particular tensors with countable sets. This reduces the claim to thenatural isomorphism 1 × X ∼ = X .The next lemma will be used, sometimes implicitly, in Section 4. Lemma 2.4. If E is countably lextensive, then the functor S S from countable sets to E preservesfinite limits.Proof. The functor S S preserves terminal objects by definition. It also preserves pullbacks.Indeed, every pullback diagram of (countable) sets decomposes as a (countable) coproduct of prod-uct diagrams. These products are preserved since products preserve countable coproducts in eachvariable by lextensivity.The next lemma will be applied in Section 6. Lemma 2.5.
Let E be an α -lextensive category. If D is a small category and S : D → Set is afunctor which takes values in α -small sets, then there is an equivalence of categories E D ↓ S ≃ E D ↓ S where D ↓ S denotes the category of elements of S .Proof. The proof is similar to that of Lemma 2.2. The functor E D ↓ S → E D ↓ S is defined bysending X → S to the diagram consisting of pullbacks of X to D ( d, − ) along all x : D ( d, − ) → S .It has a left adjoint given by coproducts over S d for all d ∈ D . The counit of this adjunctionis an isomorphism by universality of coproducts and the unit is an isomorphism by effectivity ofcoproducts. Lemma 2.6.
Let D be a small category. Let Y • : C → E D be a diagram such that Y • ( d ) admits avan Kampen colimit in E for all d ∈ D . Then Y • has a van Kampen colimit in E D . roof. If each d ∈ D , colim c ∈ C Y c ( d ) exists in E , then it is functorial in d and it is a colimit in E D .In particular, an object over colim c Y c is a D -indexed diagram X ( d ) → colim C Y c ( d ), which as thesecolimits are all van Kampen is the same as a ( C × D )-indexed diagram X c ( d ) → Y c ( d ) which isCartesian in the C -direction, which in turn is the same as a C -indexed diagram X • ∈ E D which isCartesian over Y • , hence proving the lemma.We now recall the definition of various kinds of lextensive categories [CLW93]. Definition 2.7.
Let E be a category with finite limits. For a regular cardinal α , we say that E is α -lextensive if α -coproducts exist and are van Kampen colimits. Furthermore, we say that E is(i) lextensive if it is ω -lextensive, i.e., finite coproducts exist and are van Kampen colimits,(ii) countably lextensive if it is ω -lextensive, i.e., countable coproducts exist and are van Kampencolimits,(iii) completely lextensive if it is α -lextensive for all α , i.e., all small coproducts exist and are vanKampen colimits. Example 2.8.
There are numerous examples of lextensive categories.(i) Any presheaf category is completely lextensive. In particular, for any group G the categoryof G -sets is countably lextensive.(ii) More generally, any Grothendieck topos is completely lextensive. In fact, Giraud’s theoremcharacterises Grothendieck toposes as the locally presentable categories in which coproductsand (in an appropriate sense) quotients by equivalence relations are van Kampen colimits.(iii) The category of topological spaces is completely lextensive. The same is true for many ofits subcategories such as categories of Hausdorff spaces, compactly generated spaces, weaklyHausdorff compactly generated spaces, etc.(iv) The category of affine schemes is lextensive, the category of schemes is completely lextensive.(v) The category of countable sets is countably lextensive.(vi) A category with finite limits E has the free coproduct completion which can be constructed asthe category Fam E of families of objects in E . Explicitly, an object is pair ( S, ( X s ) s ∈ S ) where S is a set and ( X s ) s ∈ S is an S -indexed family of objects of E . A morphism ( S, ( X s )) → ( S ′ , ( X ′ s ′ ))consists of a function f : S → S ′ and morphisms X s → X ′ f ( s ) for all s ∈ S . Fam E is completelylextensive. The α -coproduct completion, Fam α E , obtained by restricting to α -small families,is an α -lextensive category.We now turn our attention to the class of complemented inclusions. These will be useful forconstruction of certain colimits whose existence is not immediately obvious in lextensive categoriesand, especially, in their diagram categories. First of all, recall that a morphism i : A → B in E is a complemented inclusion if it has a complement , i.e., a morphism j : C → B such that i and j exhibit B as a coproduct of A and C in E . In other words, i is isomorphic to the coproductinclusion A → A ⊔ C . We will often say simply that C is a complement of A . The notation A B will be sometimes used to indicate complemented inclusions. Note that complemented inclusionsare sometimes (e.g., in our previous work [GSS19, Hen19]) called decidable inclusions in referenceto the notion of decidability in constructive logic.11 emma 2.9. (i) If E is lextensive, then the pushout of a complemented inclusion along arbitrary morphismexists and is again a complemented inclusion. Moreover, such pushouts are preserved byfunctors that preserve finite coproducts and are van Kampen colimits. (ii) If E is countably lextensive, then the colimit of a sequence of complemented inclusions existsand is again a complemented inclusion. Moreover, such colimits are preserved by functorsthat preserve countable coproducts and are van Kampen colimits.Proof. If i : A → B is a complemented inclusion with complement C , then the pushout of i along A → D is A ⊔ D . Similarly, if i k : A k → A k +1 are complemented inclusions with complements C k +1 ,then colim k A k is ` k C k (where C = A ). The claims on preservation then follow immediately.Since these preservation properties hold also in the evident bicategorical sense, these colimits arevan Kampen by Lemma 2.2. Lemma 2.10.
Assume E is lextensive. (i) complemented subobjects in E are closed under finite unions. (ii) complemented inclusions in E are closed under finite limits, i.e., if X → Y is a naturaltransformation between finite diagrams in E that is a levelwise complemented inclusion, thenso is the induced morphism lim X → lim Y .Proof. The proof of [GSS19, Lemma 1.1.4] applies verbatim.
Lemma 2.11.
Assume that E is countably lextensive. Then the full subcategory of [ ω, E ] consistingof sequences of complemented inclusions has finite limits which are preserved by the colimit functor(sending each sequence to its colimit in E ).Proof. First note that the category of sequences of complemented inclusions has finite limits bypart (ii) of Lemma 2.10. Moreover, part (ii) of Lemma 2.9 implies that colimits of such sequencesexist. It suffices to show that this colimit functor preserves terminal objects and pullbacks. Terminalobjects are preserved since ω is a connected category (it has an initial object). For the case ofpullbacks, we consider a span A → C ← B of sequences of complemented inclusions. We need toshow that the map colim k ∈ ω A k × C k B k → colim A × colim C colim B is invertible. We decompose this map into three factors:colim k ∈ ω A k × C k B k colim A × colim C colim B .colim k ∈ ω A k × colim C B k colim i,j ∈ ω A i × colim C B j The left map is invertible even before taking colimits because C k → colim C is a monomorphism.The bottom map is invertible because the diagonal functor ω → ω × ω is final (it has a left adjoint).The right map is invertible by universality of the van Kampen colimits colim A and colim B (part (ii)of Lemma 2.9). 12et D be a small category. We say that a morphism ϕ : F → G in E D , is a levelwise complementedinclusion if its components ϕ d : F d → G d , for d ∈ D , are complemented inclusions in E . Note thatthis is considerably less restrictive than asking for ϕ to be a complemented inclusion in E D . Corollary 2.12.
Let D be a small category. (i) If E is lextensive, then pushouts along levelwise complemented inclusions exist, are computedlevelwise and are van Kampen colimits in E D . (ii) If E is countably lextensive, then colimits of sequences of levelwise complemented inclusionsexist, are computed levelwise and are van Kampen colimits in E D .Proof. This follows immediately from Lemmas 2.6 and 2.9.
Lemma 2.13.
Let D be a small category. If E is lextensive, then the pushout products of levelwisecomplemented inclusions in E D with arbitrary morphisms exist. Moreover, the pushouts involvedare van Kampen.Proof. By universality of coproducts, levelwise complemented inclusions are closed under pullbacks.Thus a pushout computing a pushout product with a levelwise complemented inclusion is a pushoutalong a levelwise complemented inclusion. They are van Kampen by Corollary 2.12.The following statement will be needed in Section 4 to prove Lemma 4.5.
Lemma 2.14.
Let C be a category, P a poset with binary meets, X ∈ C an object and A = ( A p ֒ → X | p ∈ P ) a diagram of subobjects of X closed under intersection, i.e., such that A p ∩ A q = A p ∩ q . Then if A has a van Kampen colimit, the colimit is also a subobject of X .Proof. We assume that colim p ∈ P A p exists and is a van Kampen colimit, and we show that thediagonal map colim p ∈ P A p → F = (colim p ∈ P A p ) × X (colim p ∈ P A p ) is an isomorphism. First, weform pullbacks: A p ∩ A q F q A q F p F colim q A q A p colim p A p X Using that the colimits are van Kampen, we have that F = colim p F p and F p = colim q A q ∩ A p and hence F = colim p,q A p ∩ A q with the two maps F → colim p A p being induced by the maps A p ∩ A q → A p and A p ∩ A q → A q . We conclude by observing that colim p ( A p ∩ A q ) = A q . Indeedthe map P → ( ↓ q ) that send p ∈ P to p ∩ q is right adjoint to the inclusion of ( ↓ q ) to P , so it is afinal functor. It hence follows that colim p ∈ P A p ∩ q = colim p q A p = A q
13o this implies that F = colim q A q , with the projection map F → colim q A q being the identity,hence proving that colim q A q → X is a monomorphism.We prove a statement relating van Kampen colimits and the pullback evaluation b ev functor,defined in (1.5). This statement will be needed in Section 8. Lemma 2.15.
Let D be a small category. Let Y : C → [ D op , E ] be a diagram with levelwise vanKampen colimit colim Y . Let p : X → Y be a Cartesian transformation, which we regard as a C -indexed diagram of arrows in [ D op , E ] .Let q : A → B be a map in [ D op , Set ] with B representable such that [ D op , E ] supports evaluationat A . Then b ev q (valued in arrows of E ) preserves the colimit of p , the resulting colimit is computedseparately on source and target, and all maps of the colimit cocone are pullback squares.Proof. First note that by levelwise effectivity of colim Y , we obtain colim X (and hence colim p ).The square p c → colim p is a pullback for all c ∈ C .Consider the functor F sending an arrow M → N in [ D op , E ] to the sequence of arrows M ( B ) M ( A ) × N ( A ) N ( B ) N ( B ).The first arrow is the pullback evaluation at q of M → N . Evaluation preserves limits, in particularpullbacks. By pullback pasting, the action of F on a map of arrows that is a pullback is a pastingof pullback squares.Let us inspect the action of F on the colimit cocone of p . It will suffice to show that it resultsin objectwise colimit cocones. Since the maps of the colimit cocone of p are pullback squares, weobtain pastings of pullback squares upon applying F . Recall that ev B is computed by evaluationat the object representing B . So by assumption, (colim Y )( B ) = ev B (colim Y ) is colimit of ev B ◦ Y and van Kampen. The claim follows by universality of this van Kampen colimit. The goal of this section is to develop a version of the small object argument that allows usto construct weak factorisation systems on the category of simplicial objects s E , where E is acountably lextensive category. In view of our application to both simplicial objects in Section 4 andsemisimplicial objects in Section 12, we develop our small object argument for diagram categories E D in general. Importantly, our weak factorisation systems are enriched , in the sense of [Rie14]. Wewill be constructing Psh E -enriched weak factorisation systems on E D , where where Psh E denotesthe category of presheaves over E . This is because the category of diagrams E D is not necessarily E -enriched, but it is Psh E -enriched, as we now recall.For E ∈ E and X ∈ E D , we define E × X ∈ E D by letting( E × X ) d = def E × X d . (3.1)Given X, Y ∈ E D , we then define the hom-object Hom Psh E ( X, Y ) ∈ Psh E by letting:Hom Psh E ( X, Y ) : E op → Set E Hom
Set ( E × X, Y )14his makes E D into a Psh E -enriched category, so that the formula in (3.1) provides the tensor of E ∈ Psh E and X ∈ E D with respect to this enrichment. When the presheaf is representable, therepresenting object is denoted by Hom E ( X, Y ).Using the enrichment, we can define an internal version of the familiar lifting problems involvedin the definition of a weak factorisation systems. For morphisms i : A → B and p : X → Y in E D ,we define the presheaf of lifting problems of i against p by lettingProb Psh E ( i, p ) = def Hom
Psh E ( A, X ) × Hom
Psh E ( A,Y ) Hom
Psh E ( B, Y ) . When the relevant hom-objects are representable, then so is Prob
Psh E ( i, p ). In this case, we writeProb E ( i, p ) for its representing object and call it the object of lifting problems of i against p . Notethat the induced pullback-hom morphism of i and p (cf. Remark 1.2) has the form [ Hom
Psh E ( i, p ) : Hom Psh E ( B, X ) → Prob
Psh E ( i, p ) (3.2)Again, if the objects are representables, we have also an induced pullback-hom morphism in E ,which has the form [ Hom E ( i, p ) : Hom E ( B, X ) → Prob E ( i, p ) . (3.3)We are ready to define the Psh E -enriched counterparts of the standard lifting properties. Definition 3.1.
Let i : A → B and p : X → Y be morphisms of E D . • We say that i has the Psh E -enriched left lifting property with respect to p and that p has the Psh E -enriched right lifting property with respect to i if the induced pullback hom morphismin (3.2) is a split epimorphism in Psh E . • We say that i has the E -enriched left lifting property with respect to p and that p has the E -enriched right lifting property with respect to i if the induced pullback hom morphism in (3.3)exists and is a split epimorphism in E .Since the Yoneda embedding is fully faithful and preserves pullbacks, as soon as all relevant E -valued hom-objects exist, the Psh E -enriched left lifting property and E -enriched left lifting property are equivalent and Prob Psh E ( i, p ) is represented by Prob E ( i, p ).As is usual, we extend the terminology of enriched lifting properties from maps to classes ofmaps on either side by universal quantification. Definition 3.2.
Let I = { i : A i → B i } be a set of morphisms of E D . • An (enriched) I -fibration is a morphism with the enriched right lifting property with respectto I . • An (enriched) I -cofibration is a morphism with the enriched left lifting property with respectto I -fibrations.When the left map of a Psh E -enriched lifting problem comes from Set D via levelwise applicationof the operation in (2.2), we may simplify the lifting problem (assuming some technical conditionshold). Indeed, the pullback hom (3.3) reduces to a pullback evaluation. We record this in the nextcouple of statements, which are phrased using D op instead of D in order to exploit the language of15epresentable functors. We make use of the evaluation functor ev K : [ D op , E ] → E defined for finitecolimits K of representables by letting:ev K ( X ) = R d ∈ D op X K d d . This generalises the evaluation functor defined in Eq. (1.4), which is the case D = ∆. As inRemark 1.1, we may equivalently view ev K ( X ) as the K -weighted limit of X , which implies thatev is a (partial) two-variable functor. Lemma 3.3.
Let K ∈ [ D op , Set ] be levelwise countable. (i) There is an isomorphism ( E × K ) d ∼ = K d · E natural in K , E ∈ E , and d ∈ D . (ii) Assume that K is a finite colimit of representables. Then the hom-presheaf Hom
Psh E ( K, X ) isrepresentable for X ∈ [ D op , E ] and we have an isomorphism Hom E ( K, X ) ∼ = ev K ( X ) , naturalin K and X ∈ [ D op , E ] .Proof. Part (i) follows from Lemma 2.3. For part (ii), part (i) implies that Hom
Psh E ( K, X ) isnaturally isomorphic to the E -presheaf E Hom
Set ( d K d · E, X ). A representing object for itis by definition the K -weighted limit of X , i.e., ev K ( X ). This exists in our setting for K a finitecolimit of representables. Proposition 3.4.
Let i : A → B be a map in [ D op , Set ] between objects that are levelwise countableand finite colimits of representables and let p : X → Y be a map in [ D op , E ] . Then the following areequivalent: (i) i : A → B has the E -enriched left lifting property with respect to p , (ii) the pullback evaluation b ev i ( p ) is a split epimorphism in E .Proof. This is an immediate consequence of part (ii) of Lemma 3.3.Proposition 3.4 will be used in Section 4 to relate (trivial) Kan fibrations in s E in the sense ofDefinition 1.3 with fibrations in the sense of Definition 3.2 with respect to the images in s E of horninclusions (boundary inclusions, respectively) under the operation ( − ) : Set → E .We now turn our attention to
Psh E -enriched weak factorisation systems. Definition 3.5. A Psh E -enriched weak factorisation system on E D is a pair ( L , R ) of classes ofmorphisms of E D such that: • a morphism belongs to L if and only if it has the Psh E -enriched left lifting property withrespect to R ; • a morphism belongs to R if and only if it has the Psh E -enriched right lifting property withrespect to L ; • every morphism of E D factors as an L -morphism followed by an R -morphism.We will abbreviate “ Psh E -enriched lifting property” to “enriched lifting property”, but we willbe explicit about cases where it coincides with the E -enriched lifting property. Lemma 3.6.
Let ( L , R ) be an enriched weak factorisation system. A morphism is in L if and only if it has the ordinary left lifting property with respect to R . (ii) A morphism is in R if and only if it has the ordinary right lifting property with respect to L .In particular, ( L , R ) is also an ordinary weak factorisation system.Proof. For (i), a morphism of L has the ordinary left lifting property with respect to R by eval-uating the hom-presheaves at 1 ∈ E . Conversely, a morphism with such a lifting property is in L by the retract argument as in the proof of Proposition 3.17 below. Part (ii) follows by duality.We will fix a set I and study a version of the small object argument that produces an enrichedweak factorisation system of I -cofibrations and I -fibrations under suitable assumptions. Definition 3.7.
Let i : A → B and p : X → Y be morphisms of E D . Assume that we have afactorisation X Y . X ′ p p ′ We say that p satisfies the X ′ -partial enriched right lifting property with respect to i if there is alift in the diagram Hom Psh E ( B, X ′ )Prob Psh E ( i, p ) Prob Psh E ( i, p ′ ).Such partial lifting properties are a crucial ingredient of the small object argument, but theyare only tractable when i is a levelwise complemented inclusion. This is thanks to the next twolemmas, where we use the tensor defined in (3.1). Lemma 3.8.
Levelwise complemented inclusions in E D are closed under: (i) E × − for all E ∈ E ; (ii) countable coproducts; (iii) pushouts along arbitrary morphisms; (iv) sequential colimits; (v) retracts.Moreover, the colimits of parts (ii), (iii) and (iv) are preserved by E × − for all E ∈ E .Proof. The functor E × − and all the colimits mentioned are computed levelwise in E , so theresults boil down to the fact that complemented inclusions in E are stable under all these con-structions. Stability under E × − follows from distributivity of product over coproduct in com-plemented categories: if A → A ⊔ B is a complemented inclusion, then its image under E × − is E × A → ( E × A ) ⊔ ( E × B ) and is a complemented inclusion. The case of a countable coproduct17s also clear: if A k → A k ⊔ B k is a family of complemented inclusions, then their coproduct can bewritten as ` A k → ( ` A k ) ⊔ ( ` B k ). Stability under pushout and sequential composition followsfrom Lemma 2.9. The fact that they are preserved by E × − follows from Lemma 2.9. The case ofretracts can be deduced from the stability under limits proved in Lemma 2.10 as retracts can beseen as limits. Lemma 3.9.
Let p : X → Y be a map in E D and L a class of levelwise complemented inclusionsin E D that have the enriched left lifting property with respect to p . Then L is closed under thefollowing operations: (i) tensors by objects of E , (ii) countable coproducts, (iii) pushouts, (iv) colimits of sequences, (v) retracts.Proof. For X ∈ E D , the functor Hom Psh E ( − , X ) is not necessarily an adjoint. However, since splitepimorphisms are closed under limits, it is sufficient to verify that it carries the relevant colimitsto limits. (In the case of tensors this means that Hom Psh E ( F × A, X ) ∼ = Hom Psh E ( A, X ) E ( − ,F ) forall F ∈ E .) This follows directly from these colimits being preserved by the tensors as recorded inLemma 3.8. Definition 3.10.
Let A ∈ E D . We say that A is finite if the following hold:(i) Hom E ( A, X ) exists for every X ∈ E D ;(ii) Hom E ( A, − ) preserves colimits of sequences of levelwise complemented inclusions;(iii) Hom E ( A, − ) sends levelwise complemented inclusions to complemented inclusions.The next lemma provides a supply of finite objects. For its statement, recall the functor S S from Section 2. As Lemma 3.3, it is formulated using D op instead of D for convenience. Lemma 3.11.
Let D be a locally countable category and assume that presheaf A ∈ Psh D is a finitecolimit of representables. Then A ∈ [ D op , E ] is finite.Proof. First, note that since D is locally countable, A is levelwise countable and thus A exists. Bypart (ii) of Lemma 3.3, Hom E ( A, − ) exists and is given by ev A (evaluation at A ). Call X ∈ Psh D E -finite if it satisfies the conditions of Definition 3.10 with Hom E ( X, − ) replaced by ev X . Our goalthen is to show that A is E -finite. This follows from the following observations: • Representables are E -finite. For this, recall that evaluation at a representable is given byevaluation at the representing object. Part (ii) uses part (ii) of Corollary 2.12 to see that thecolimit is computed levelwise. 18 E -finite presheaves are closed under finite colimits. For this, we use that the partial two-variable functor ev sends colimits in its first argument to limits. Part (i) holds since E hasfinite limits. Part (ii) holds since finite limits preserve colimits of sequences of complementedinclusions in E (Lemma 2.11). Part (iii) holds since complemented inclusions in E are closedunder finite limits (part (ii) of Lemma 2.10).The hypothesis of finiteness is used in the next result, where we use the notion of an I -fibrationin the sense of Definition 3.2. Lemma 3.12.
Assume that the domains and codomains of morphisms of I are finite. Let Y ∈ E D and ( X k → X k +1 | k ∈ N ) be a sequence of morphisms in E D ↓ Y . If every X k → X k +1 is a levelwisecomplemented inclusion and each p k : X k → Y has X k +1 -partial enriched right lifting property withrespect to I , then colim k X k → Y is an I -fibration.Proof. Fix a morphism i : A → B of I . Since A and B are finite, the given partial enriched liftingproperties are E -enriched. Moreover, since X k → X k +1 is a levelwise complemented inclusion,Lemma 2.10 implies that Prob E ( i, p k ) → Prob E ( i, p k +1 ) is a complemented inclusion.Proceeding by induction with respect to k , we can pick liftsHom E ( B, X k +1 )Prob E ( i, p k ) Prob E ( i, p k +1 )that are natural in k . Indeed, since Prob E ( i, p k − ) → Prob E ( i, p k ) is a complemented inclusion, wecan construct a compatible lift by assembling a previously constructed lift on Prob E ( i, p k − ) witha given lift on its complement. Since A and B are finite, we havecolim k Hom E ( B, X k ) = Hom E ( B, colim k X k )and colim k Prob E ( i, p k ) = colim k (cid:0) Hom E ( A, X k ) × Hom E ( A,Y ) Hom E ( B, Y ) (cid:1) = (cid:18) colim k Hom E ( A, X k ) (cid:19) × Hom E ( A,Y ) Hom E ( B, Y )= Hom E ( A, colim k X k ) × Hom E ( A,Y ) Hom E ( B, Y )= Prob E ( i, colim k p k ),the latter by universality of sequential colimits of complemented inclusions in E (Lemma 2.9). Thuswe obtain a diagram Hom E ( B i , colim k X k )Prob E ( i, colim k p k ) Prob E ( i, colim k p k ).19here the bottom map is an identity, i.e., these lifts form a section that exhibits colim k X k → Y as an I -fibration.The following lemma isolates a simpler version of the inductive step in the construction of liftsin Lemma 3.12. It is needed in Section 8. Lemma 3.13.
Let
X YA B . p q be a pullback square in E D with A → B a levelwise complemented inclusion. Let i : U → V be amap in E D between finite objects such that [ Hom E ( i, p ) and [ Hom E ( i, q ) have sections. Then, for anysection s of [ Hom E ( i, p ) , there is a section t of [ Hom E ( i, q ) such that the diagram Hom E ( V, X ) Hom E ( V, Y )Prob E ( i, p ) Prob E ( i, q ) . [ Hom E ( i, p ) [ Hom E ( i, p ) forms a morphism of retracts.Proof. The map Prob E ( i, p ) → Prob E ( i, q ) is a complemented inclusion by Lemma 2.10. We con-struct t by using s on Prob E ( i, p ) and a given section on its complement. Theorem 3.14 (Enriched small object argument) . Let I = ( i : A i → B i | i ∈ I ) be a countableset of levelwise complemented inclusions between finite objects of E D . Then I -cofibrations and I -fibrations form an enriched weak factorisation system in E D .Proof. For a morphism p : X → Y we form a sequence X → X → X → . . . in E ↓ Y byiteratively taking pushouts ` i ∈ I Prob E ( i, p k ) × A i X k ` i ∈ I Prob E ( i, p k ) × B i X k +1 Y . p k p k +1 The adjoint transpose of Prob( i, p k ) × B i → X k +1 witnesses the X k +1 -partial enriched right liftingproperty of p k with respect to i . Moreover, by Lemma 3.8, X k → X k +1 is a levelwise complementedinclusion. Thus Lemma 3.12 applies and shows that colim k X k → Y is an I -fibration. UsingLemma 3.9, we show that X → colim k X k is an I -cofibration.20 emark 3.15. Essentially the the same argument used to prove Theorem 3.14 can be used toprove a more general statement. Namely, instead of E D we consider an E -module C , i.e., a categoryequipped with a tensor functor − × = : E × C → C that is associative in the sense that the functor
E →
End C , given by E ( E × − ), is monoidal (with respect to the Cartesian product on E andfunctor composition on End C ). Then C carries a Psh E -enrichment defined in the same way as theone on E J which yields notions of an enriched lifting property and an enriched weak factorisationsystem. The complication lies in the fact that the definition of levelwise complemented inclusions isnot available in C . However, if we assume that C is equipped with a class of morphisms D satisfyingthe conclusion of Lemma 3.8, then the proof of Theorem 3.14 applies without changes. (Note thatin this case the notion of finiteness in C depends on the choice of D .) Examples of categoriesthat can be endowed with such structure include the categories of internal categories in E , internalgroupoids in E and marked simplicial objects in E .We conclude this section by introducing the notion of a cell complex and establish a few resultsthat will be useful later. Definition 3.16.
For a family of maps I = ( i : A i → B i | i ∈ I ), an I -cell complex is a morphismof E D that is a sequential colimit of maps X → Y arising as pushouts ` i E i × A i X ` i E i × B i Y for some family ( E i ) i ∈ I of objects of E . Proposition 3.17.
Under the hypotheses of Theorem 3.14, a morphism of E D is an I -cofibrationif and only if it is a codomain retract of an I -cell complex. In particular, every I -cofibration is alevelwise complemented inclusion.Proof. A retract of an I -cell complex is an I -cofibration by Lemma 3.9. It is furthermore a levelwisecomplemented inclusion by Lemma 3.8. Conversely, let X → Y be an I -cofibration and considerthe factorisation X → X ′ → Y defined in the proof of Theorem 3.14. Then X → X ′ is an I -cell complex by construction. Moreover, X → Y has the Psh E -enriched left lifting property withrespect to X ′ → Y and, in particular, it has the ordinary left lifting property (by evaluating thehom-presheaves at the terminal object). Thus there is a lift in the diagram X X ′ Y Y which exhibits X → Y as a codomain retract of X → X ′ . Lemma 3.18.
In the setting of Theorem 3.14, the following hold. (i)
Consider a countable family of maps f k in the arrow category of E D . If f k is an I -fibration forall k , then so is the coproduct ` k f . When E is α -lextensive, the same holds for α -coproducts. Consider a span f ← f → f in the arrow category of E D . Assume that both legs formpullback squares and that f → f is a levelwise complemented inclusion on codomains. If f k is an I -fibration for k = 0 , , , then so is the pushout colim f . (iii) Consider a sequential diagram f → f → . . . in the arrow category of E D . Assume thatthe maps f k → f k +1 form pullback squares and are levelwise complemented inclusions oncodomains. If f k is an I -fibration for all i , then so is colim f .Proof. In all three parts, the colimit colim f exists and is computed separately on sources andtargets where they form van Kampen colimits by Corollary 2.12. Let C denote the shape of thediagram (which varies over the parts). We check that colim f is an I -fibration using Proposition 3.4.For each i ∈ I , given a section of b ev i ( f c ) for c ∈ C , we have to construct a section of b ev i (colim f ).Using Lemma 2.15 and functoriality of colimits, it suffices to construct a family of section of b ev i ( f c )that is natural in c ∈ C .For part (i), the naturality is vacuous. For part (ii), we pull the section of b ev i ( f ) back to asection of b ev i ( f ) and then use Lemma 3.13 to replace the section of b ev i ( f ) by one that cohereswith the one of b ev i ( f ). For part (iii), we recurse on k and use Lemma 3.13 to replace the givensection of b ev i ( f k +1 ) by one that coheres with the one of b ev i ( f k ). In all three cases, the sectionsform a D -shaped natural transformation as required.We consider the application functor app : [ C , D ] × C → D and record some commonly used factsabout pushout applications in the following statement. We regard the pushout application of anatural transformation [ C , D ] to an arrow in C to be defined if the pushout in the evident commutingsquare exists. Recall that the pushout application is the induced arrow from the pushout corner. Lemma 3.19.
Let u : X → Y be a map in [ C , D ] . Then pushout application d app( u, − ) : C [1] → D [1] forms a partial functor with the following properties. (i) Let c : I → C [1] be a diagram of arrows with levelwise colimit (i.e., a colimit that is com-puted separately on sources and targets in C ). If X and Y preserve this levelwise colimit and d app( u, − ) is defined on all values of c , then d app( u, − ) preserves the levelwise colimit of c . (ii) Let f → g be a morphism in C [1] that is a pushout square. If X and Y preserve this pushoutand d app( u, − ) is defined on f and g , then d app( u, f ) → d app( u, g ) is a pushout square. (iii) For an ordinal α , let A → A → . . . → A α be an α -composition in C . If this α -composition ispreserved by X and Y and d app( u, − ) is defined on A β → A β ′ for β ≤ β ′ ≤ α , then d app( u, − ) preserves the given the α -composition and the resulting step map at β < α is a pushout of d app( u, − ) applied to A β → A β +1 .Proof. This is folklore technique in abstract homotopy theory. Similar proofs (in a slightly differentcontext) can be found in [RV14, Sections 4 and 5], in particular [RV14, Lemma 4.8] for part (i)and [RV14, Lemma 5.7] for parts (ii) and (iii).
Lemma 3.20.
Let
F, G : E D → E D ′ be two functors that preserves levelwise complemented maps,their pushouts and their sequential compositions. We assume that F and G are equipped withisomorphisms F ( E × X ) ∼ = E × F ( X ) G ( E × X ) ∼ = E × G ( X )22 atural in E ∈ E and X ∈ E D (respectively, X ∈ E D ′ ) and let λ : F → G be a natural transformationcompatible with these isomorphisms. Let I D ⊆ ( E D ) [1] and I D ′ ⊆ ( E D ′ ) [1] be countable sets ofarrows satisfying the conditions of Theorem 3.14. If for each i ∈ I D , the pushout application d app( λ, i ) is an I D ′ -cofibration, then for each I D -cofibration i , the pushout application d app( λ, i ) isan I D ′ -cofibration.Proof. First, because of Lemma 3.8, all I D -cofibrations are levelwise complemented inclusions, sotheir image under F are again levelwise complemented inclusions and hence pushouts along themexist. This shows that d app( λ, i ) always exists when i is an I D -cofibration.By Proposition 3.17, a general a I D -cofibration is a retract of a sequential composite of pushoutsof countable coproducts of the form E × A → E × B for a map A → B in I D and E ∈ E . A map E × i : E × A → E × B is sent by d app( λ, − ) to the map E × d app( λ, i ), so as we are assuming thatfor each i ∈ I D the map d app( λ, i ) is an I D ′ -cofibration, it follows that the map of the form E × i are also sent to I D ′ -cofibration.Using Lemma 3.19 one concludes that any transfinite composition of pushouts of maps of theform E × i for i ∈ I D is also sent by d app( λ, − ) to a I D ′ -cofibration. Finally, as d app( λ, − ) is a functorit preserves retract, and so retracts of such maps are also sent to I D ′ -cofibration, and this concludesthe proof as any I D -cofibration is a retract of such a transfinite composition of pushouts. Proposition 3.21.
Let j : X → Y be a morphism of E D . Under the hypothesis of Theorem 3.14,if i b × j is an I -cofibration for all i ∈ I , then f b × j is an I -cofibration for all I -cofibrations f .Proof. We apply Lemma 3.20 to the natural transformation − × j : − × X → − × Y of endofunctorson E D . Let us check the needed preservation properties of the endofunctor − × Z on E D for Z ∈E . Preservation of levelwise complemented inclusions follows from preservation of complementedinclusions in E under product with a fixed object (a consequence of lextensivity). Preservation ofthe relevant colimits involving levelwise complemented inclusions is an instance of Corollary 2.12.Preservation of tensors with objects of E reduces to associativity and commutativity of productsin E ; this is natural, so the map − × j : − × X → − × Y respects the witnessing isomorphism asappropriate. In this section we consider a countably lextensive category E . We construct two weak factori-sation systems on the category s E of simplicial objects in E that will be proven to form a modelstructure in Section 9. Our main goal is to describe the resulting cofibrations in Theorem 4.6 whichrelies on identification of one of the factorisation systems as a Reedy factorisation system (Proposi-tion 4.3). In our setting, the category s E has relatively few colimits and consequently much of thissection is committed to discussion of the Reedy theory under these weak hypotheses.We will use the enriched small object argument of Theorem 3.14 with the generating sets ob-tained by applying the partial functor of (2.2) to the sets of boundary inclusions and horn inclusionsin (1.6), i.e., I s E = { ∂ ∆[ n ] → ∆[ n ] | n ≥ } and J s E = { Λ k [ n ] → ∆[ n ] | n ≥ k ≥ , n > } .We will refer to ∆[ m ] as a simplex in s E and similarly for boundaries and horns. We say that amap in s E is a cofibration if it is a I s E -cofibration and that it is a trivial cofibration if it is a J s E -cofibration. Moreover, we note that notions of (Kan) fibrations and trivial (Kan) fibrations asintroduced in Definition 1.3 coincide with the notions of J s E -fibrations and I s E -fibration.23 roposition 4.1. Let f : X → Y be a map in s E . (i) f is a fibration if and only if it is a J s E -fibration; (ii) f is a trivial fibration if and only if it is a I s E -fibration.Proof. By Proposition 3.4, the condition of Definition 1.3 for f being a (trivial) Kan fibration isequivalent to the E -enriched right lifting property of f with respect to J s E (respectively, I s E ).The existence of weak factorisation systems linking these classes is a direct consequence of theresults of Section 3. Theorem 4.2.
Let E be a countably lextensive category. The category s E of simplicial objects in E admits two weak factorisation systems: • cofibrations and trivial fibrations, cofibrantly generated by I s E ; • trivial cofibrations and fibrations, cofibrantly generated by J s E .Proof. All morphisms of I s E and J s E are levelwise complemented inclusions since S S preservescomplemented inclusions. Moreover, their domains and codomains are finite colimits of representa-bles and thus Lemma 3.11 implies that the assumptions of Theorem 3.14 are satisfied.Recall that E admits a weak factorisation system consisting of complemented inclusions as leftmaps and split epimorphisms as right maps. We now wish to characterise our cofibrations andtrivial fibrations in terms of the induced Reedy weak factorisation on s E . Traditional treatmentsof Reedy theory such as [RV14] tacitly assume that the underlying category is bicomplete; this isnot the case here. Separately, there is the treatment [RB06] of Reedy theory in the context of a(co)fibration category, but it only considers Reedy left or right maps between Reedy left or rightobjects; in our setting, not all objects are Reedy cofibrant or fibrant. Let us thus discuss some ofthe details of the Reedy weak factorisation system on s E .Let m ≥
0. We write ∆ op [ m ] for ∆([ m ] , − ), i.e., the functor in [∆ , Set ] corepresented by m . The coboundary ∂ ∆ op [ m ] of ∆ at level m is the subobject of ∆ op [ m ] consisting of those maps whichare not face maps. Equivalently, ∂ ∆ op [ m ] k ⊆ ∆([ m ] , [ k ]) consists of those maps [ m ] → [ k ] whosedegeneracy-face factorisation has non-identity degeneracy map.Let A ∈ s E . The latching object L m A , if it exists, is the colimit of A weighted by ∂ ∆ op [ m ]. Wehave a canonical map L m A → A .Let i : A → B be a map in s E and m ≥
0. We wish to consider the relative latching mapof i . Ordinarily, we would define it as the map A m ⊔ L m A L m B → B m . However, its domaindepends on the existence of the latching objects L m A and L m B and a pushout. We wish to avoidthese assumptions. Consider the functor s E →
Set sending X to the set of pairs consisting of amap u : A m → X and a natural family v f : B k → X for f : [ m ] → [ k ] not a face map such that u ◦ Af = v f ◦ i k . If this functor has a corepresenting object, we denote it by A m ⊔ L m A L m B andobtain the relative latching map A m ⊔ L m A L m B → B m of i at level m . If L m A and L m B exist, thisagrees with the description in terms of the pushout suggested by our notation.We desire a more abstract view on the relative latching map. For this, we introduce the notionof pushout weighted colimit. Consider the two-variable functor H : [∆ , Set ] op × s E op → [ E , Set ] (4.1)24ending W and X to I [∆ , Set ]( W, E ( X ( − ) , I )) Recall that a W -weighted colimit of X , denotedcolim W X , is by definition a representing object of H ( W, X ). The pullback construction of H is thetwo-variable functor b H : ([∆ , Set ] op ) [1] × ( s E op ) [1] → [ E , Set ] [1] sending w : U → V in [∆ , Set ] and i : A → B in s E to the map H ( V, B ) → H ( V, A ) × H ( U,A ) H ( U, B ) (4.2)in [ E , Set ]. Assume that domain and codomain of (4.2) have representing objects Y and X , respec-tively (in particular, Y is the V -weighted colimit of B ). Then under the Yoneda embedding of E op into [ E , Set ], (4.2) corresponds to a map X → Y in E . We define this to be the pushout weightedcolimit with w : U → V of i : A → B and denote it by [ colim w i . It forms a partial two-variablefunctor [ colim ( − ) (=) : [∆ , Set ] [1] × s E [1] → [ E , Set ] [1] . Note that this is more general than a partially defined pushout construction of the two-variableweighted colimit functor because we do not require the individual colimits of A with weight V and B with weights U and V to exist.Unfolding the codomain of (4.2), we see that the relative latching map of i : A → B at level m is precisely the pushout weighted colimit of i with the coboundary inclusion ∂ ∆ op [ m ] → ∆ op [ m ].Each side exist when the other does. This point of view is useful because it enables us to obtainpushout weighted colimits of i with certain inclusions as cell complexes of relative latching maps.We call a map i a Reedy complemented inclusion if, for all m , the relative latching map of i at level m exists and is a complemented inclusion. This condition for m < k suffices to guaranteethe existence of the relative latching map at level m = k . Thus, in the inductive verification thata map is a Reedy complemented inclusion, the relevant latching maps always exist. Given a map X → Y in s E , the relative matching map at level m is its weighted limit, i.e., pullback evaluation,at ∂ ∆[ m ] → ∆[ m ], i.e., the map X m → Y m × ev ∂ ∆[ m ] Y ev ∂ ∆[ m ] X . We call X → Y a Reedy splitepimorphism if all its relative matching maps are split epimorphisms.Following standard Reedy theory, Reedy complemented inclusions and Reedy split epimorphismsform a weak factorisation system. For this, we observe that instantiating the treatment of [RV14]and making use of Lemma 3.19, the use of (co)limits in s E may be reduced to pushouts along com-plemented inclusions and pullbacks along split epimorphisms. We now relate this weak factorisationsystem to that of cofibrations and trivial fibrations, given in Theorem 4.2 (cf. also Proposition 4.1). Proposition 4.3.
The weak factorisation system of cofibrations and trivial fibrations of Theo-rem 4.2 and Proposition 4.1 coincides with the weak factorisation system of Reedy complementedinclusions and Reedy split epimorphisms.Proof.
Two weak factorisation systems coincide as soon as their right classes do. But, by inspectingthe definition of a trivial fibration in Definition 1.3, a map in s E is a Reedy split epimorphism ifand only if it is a trivial Kan fibration.The next lemma will be useful to simplify some saturation arguments in Section 6, as it allowsus to avoid considering retracts, cf. the notion of a cell complex in Definition 3.16. Lemma 4.4.
Every cofibration in s E is an I s E -cell complex. roof. If A → B is a cofibration, then B can be written as the colimit of its skeleta relative to A :Sk − A B Sk A B Sk A B . . . where Sk − A B = A and for k ≥ B k × ∂ ∆[ k ] ∪ ( A m ⊔ L m A L m B ) × ∆[ k ] Sk k − A BB k × ∆[ k ] Sk kA B is a pushout. These statements are justified analogously to the proofs of [GSS19, Lemma 2.3.1,Corollary 2.3.3]. The colimits used in the construction exist by Corollary 2.12 since they are colim-its of sequences of levelwise complemented inclusions and pushouts along levelwise complementedinclusions which is ensured by the assumption that A → B is a cofibration.Our next goal is to provide a characterisation of cofibrations in terms of actions of degeneracyoperators, stated in Theorem 4.6 below. This is a generalisation of [Hen18, Proposition 5.1.4] or[GSS19, Proposition 1.4.4] to a setting without arbitrary colimits. The proof is made significantlymore complex by the fact that E is not assumed to be a Grothendieck topos. Instead, the requiredexactness properties are substituted by Lemma 2.14. We also need the following statement. Forthis, we observe that our discussion of Reedy theory and latching objects for the case of ∆ appliesjust as well to arbitrary countable Reedy categories of countable height. Lemma 4.5.
Let D be a finite direct category. Let F : D → s E be a Reedy cofibrant diagram(including the assumption that all latching objects exist). Then the colimit of F exists and is vanKampen.Proof. We proceed by induction on the height of D . For height 0, note that D is the empty andthe claim holds because initial objects are van Kampen since s E is lextensive.Now assume the claim for height n and let D have height n + 1. Let D ′ of height n denote therestriction of D to objects of degree below n . Let I be the collection of objects of D of degree n .As per usual Reedy theory, we may compute the colimit of F as the following pushout: ` i ∈ I L i F colim D ′ F | D ′ ` i ∈ I F ( i ) colim D F .Here, the left map is a cofibration because it is a finite coproduct of cofibrations, and hence thepushout exists and is van Kampen by Lemma 2.9. By the inductive hypothesis, the colimit com-puting the latching object L i F for i ∈ I is van Kampen, and so is the colimit of F | D ′ . The finitecoproducts are van Kampen since s E is lextensive. Using the characterisation of van Kampencolimits given by Lemma 2.2, one sees that colim D F is van Kampen. Theorem 4.6 (Characterisation of cofibrations) . Let i : A → B be a map in s E . Then the followingare equivalent: the map i is a cofibration; (ii) the map i is a levelwise complemented inclusion and the map A m ⊔ A n B n → B m is a comple-mented inclusion for every degeneracy operator [ m ] _ [ n ] .Proof. We use from Proposition 4.3 that cofibrations are the same as Reedy complemented inclu-sions. As in [RV14], we work freely with pushout weighted colimits in E , with index category both ∆and its wide subcategory ∆ − of degeneracy operators. As explained above (in the case of ∆), theseare partial two-variable functors in our situation. Mirroring our notation for ∆, we write ∂ ∆ op − [ m ]for the subobject of ∆ op − [ m ] = ∆([ m ] , − ) in [∆ − , Set ] consisting of the non-identity maps. Recallthat the coboundary inclusion ∂ ∆ op [ m ] → ∆ op [ m ] arises as left Kan extension along ∆ − → ∆ ofthe coboundary inclusion ∂ ∆ op − [ m ] → ∆ op − [ m ]. For working with weighted colimits, we recall thatleft Kan extension on the side of the weight corresponds to restriction on the side of the diagram.We start with the direction from (i) to (ii). Let i be a Reedy complemented inclusion. Thenthe pushout weighted colimit of i with any finite cell complex (finite composite of pushouts) ofcoboundary inclusions is a complemented inclusion. In particular, the pushout weighted colimit ofthe restriction i | ∆ − of i to ∆ − with any finite cell complex of coboundary inclusions ∂ ∆ op − [ k ] → ∆ op [ k ] of ∆ − is a complemented inclusion. For m ≥
0, the map A m → B m is the pushout weightedcolimit of i | ∆ − with such a finite cell complex ∅ → ∆ op − [ m ], hence a complemented inclusion. Everydegeneracy operator [ m ] _ [ n ] has a section. It follows that ∆ op − [ n ] → ∆ op − [ m ] is an inclusion withlevelwise finite complement, thus we can write it as a finite cell complex of coboundary inclusionsof ∆ − . Therefore, the pushout weighted colimit of i with ∆ op − [ n ] → ∆ op − [ m ] is a complementedinclusion. But this is the map A m ⊔ A n B n → B m .We finish with the direction from (ii) to (i). We show that the relative latching map A m ⊔ L m A L m B → B m of i is a complemented inclusion by induction on m . Recall that this is the pushoutweighted limit of i | ∆ − with ∂ ∆ op − [ m ] → ∆ op − [ m ]. Let ∂ (∆ op − ↓ [ m ]) denote the opposite of the posetof non-identity degeneracy operators with source [ m ]. Consider the diagram F : ∂ (∆ op − ↓ [ m ]) →E ↓ B m sending a degeneracy operator [ m ] _ [ n ] to the object A m ⊔ A n B n . It lives canonicallyunder the object A m over B m . By switching from the weighted colimit to the conical colimit pointof view, the object A m ⊔ L m A L m B is the colimit of F in the category of factorisations of A m → B m .Equivalently, in the slice over B m , the object A m ⊔ L m A L m B is the colimit of the diagram F ∗ thatis F with shape adjoined with an initial object sent to A m .Note that, using our assumptions, we can regard F as a diagram of complemented subobjects of B m that are bounded from below by the complemented subobject A m . It remains to show that thecolimit of F ∗ in the slice over B m has a complemented inclusion as underlying map. It will sufficeto show that this colimit is subterminal. For then, it is given by the non-empty finite union of thesubobjects that constitute the values of F ∗ , and complemented subobjects are closed under finiteunions by part (i) of Lemma 2.10.The indexing category of F ∗ is a finite direct category. The latching map of F ∗ at the initialobject is 0 → A m , a complemented inclusion. The latching map of F ∗ at an object [ m ] _ [ n ] isa pushout of the relative latching map of A → B at [ m ], a complemented inclusion by inductionhypothesis. Thus, the diagram F ∗ is Reedy cofibrant. By Lemma 4.5, the colimit of F ∗ is vanKampen. All of this holds both in E as well as its slice over B m .Given a complemented subobject U → B m and an arbitrary subobject V → B m , the pushoutcorner map in the pullback of U → B m and V → B m exists. If it is a monomorphism, it computesthe union U ∪ V → B m of the given subobjects. Since the Reedy category ∆ is elegant [BR13], the27atural transformation i | ∆ − is Cartesian. This makes the value of F at an object [ m ] _ [ n ] theunion of the subobjects A m → B m and B n → B m .Using again that ∆ is elegant, given non-identity degeneracy operators [ m ] _ [ n i ] for i = 1 , m ] [ n ][ n ] [ k ]in ∆ with [ n ] _ [ k ] and [ n ] _ [ k ] degeneracy maps. Note that [ m ] _ [ k ] is distinct from theidentity. By absoluteness, we obtain a pullback B k B n B n B m .We now work in subobjects of B m . From the above pullback, we have B k = B n ∩ B n . Usingfrom Lemma 2.9 twice that pushouts along complemented inclusions are stable under pullback, wecompute ( A m ∪ B n ) ∩ ( A m ∪ B n ) = (( A m ∪ B n ) ∩ A m ) ∪ (( A m ∪ B n ) ∩ B n )= A m ∪ (( A m ∪ B n ) ∩ B n )= A m ∪ ( B n ∩ B n )= A m ∪ B k . We obtain, in subobjects of B m , that F at [ m ] _ [ n ] is the intersection (computed as pullback) of F at [ m ] → [ n ] and [ m ] → [ n ]. Thus, in subobjects of B m , the diagram F (and then also F ∗ )preserves binary meets. Recollecting from above that the colimit of F ∗ in the slice over B m is vanKampen, Lemma 2.14 shows that it is subterminal. This section is devoted to further study of weak factorisation systems constructed in Section 4,in preparation for the proof of the existence of the effective model structure. We begin with asimple verification.
Lemma 5.1. If A → B is a (trivial) cofibration between levelwise countable simplicial sets, then A → B is a (trivial) cofibration in s E .Proof. Recall that the partial functor X X is a partial left adjoint to the levelwise global sectionsfunctor. This is equivalently the functor Hom sSet (1 , − ) with 1 ∈ s E from Section 1. By adjointnessusing the weak factorisation systems of Theorem 4.2 and Proposition 4.1, it suffices to show thatHom sSet (1 , − ) preserves (trivial) fibrations. This holds by Proposition 1.4. Proposition 5.2.
Trivial fibrations are fibrations. (ii)
Trivial cofibrations are cofibrations.Proof.
The first part is immediate since trivial Kan fibrations are Kan fibrations in simplicial sets.The second parts follows by adjointness using the weak factorisation systems of Theorem 4.2.We now establish some formal properties of the two enriched weak factorisation systems, re-garding the pushout-product, pushout-tensor and pullback-cotensor functors (cf. Remark 1.2).
Proposition 5.3 (Pushout-product properties) . (i) In s E , cofibrations are closed under pushout product. (ii) In s E , the pushout product of a cofibration and a trivial cofibration is a trivial cofibration.Proof. For part (i), recall that cofibrations in sSet are closed under pushout product. Since S S preserves pushouts and products, it follows that the pushout product of generating cofibrations in s E is a cofibration. The same follows for general cofibrations in s E by Proposition 3.21. Thesepushout products exist by Lemma 2.13.For part (ii), The result holds in sSet by [GZ67, Proposition IV.2.2] and thus it carries overto s E by the argument of part (i). Lemma 5.4.
Let X ∈ s E . For every finite simplicial set K , the tensor K · X exists and is givenby K × X .Proof. Given Y ∈ s E , a morphism X → K ⋔ Y consists of a family of morphisms X m → Y ( K × ∆[ m ]) n n ,natural in m and dinatural in n . This corresponds to a family of morphisms K × ∆[ m ] n × X m → Y n ,dinatural in m and natural in n . Moreover: K × ∆[ m ] n × X m = K n × Hom([ m ] , [ n ]) × X m .Since R [ m ] Hom([ m ] , [ n ]) × X m = X n , such family of maps corresponds to a morphism K n × X n → Y n natural in n , i.e., a morphism K × X → Y in s E . Proposition 5.5 (Pushout tensor properties) . Let A → B be a cofibration between finite simplicialsets. Then, the pushout tensor with A → B exists. Furthermore, (i) it preserves trivial cofibrations, (ii) it preserves cofibrations, (iii) if A → B is a trivial cofibration, then it sends cofibrations to trivial cofibrations.Proof. The existence follows from Corollary 2.12 and Lemma 5.4. These other statements are dualto the ones of part (i) of Lemma 1.5 under the tensor-cotensor adjunction of Lemma 5.4. Notethat for this conclusion it suffices to consider the underlying ordinary weak factorisation system ofLemma 3.6 so that we do not need to verify that the adjunction is enriched over
Psh E . See [Hen18, Proposition 5.1.5] or [GSS19, Proposition 1.3.1] for the constructive version of this fact. See [Hen18, Corollary 5.2.3] or [GSS19, Proposition 1.3.1] for the constructive version of this fact.
29e now turn our attention to the cofibrations and the cofibrant objects in s E . From Section 3and Proposition 4.1 these are exactly the maps with the left lifting property with respect to Kanfibrations. The next lemma provides us with a stock of cofibrant objects. Lemma 5.6. (i)
Let E ∈ E . The constant simplicial object E ∈ s E is cofibrant. (ii) The domains and codomains of all morphisms of I s E and J s E are cofibrant. (iii) Let X ∈ s E and K be a finite simplicial set. If X is cofibrant, then so is K ⋔ X .Proof. For part (i), by Lemma 3.9, the tensor of ∂ ∆[0] → ∆[0] with E is a cofibration. ByLemma 5.4, this map is the tensor of E ∈ s E with ∂ ∆[0] → ∆[0], i.e., the map ∅ → E in s E . Part (ii) holds since S S preserves cofibrations by Lemma 5.1. Finally, for part (iii), if[ m ] _ [ n ] is a degeneracy operator, then the map ( K ⋔ X ) n → ( K ⋔ X ) m can be identified with themap X ( K × ∆[ n ]) → X ( K × ∆[ m ]). It follows from [Hen19, Proposition 3.1.11] that when K is afinite simplicial set, the map K × ∆[ n ] → K × ∆[ m ] is a finite composite of pushouts of degeneracyoperators. This implies that the map ( K ⋔ X ) n → ( K ⋔ X ) m is a finite composite of pullbacksof degeneracy map X a → X b . As X is cofibrant these maps are all complemented inclusions,hence as complemented inclusions are closed under pullback and composition, this implies that( K ⋔ X ) n → ( K ⋔ X ) m is a complemented inclusion as well. Lemma 5.7.
Cofibrations are closed under pullback along a monomorphism.Proof.
Consider a pullback square of simplicial objects: S ′ SA B
We check that S ′ → S is a cofibration using characterisation (ii) of Theorem 4.6. In an lextensivecategory, a pullback of a complemented inclusion is a complemented inclusion, hence the map S ′ → S ′ is a levelwise complemented inclusion. Given any degeneracy operator [ m ] _ [ n ], as it isa split epimorphism and S → B is a monomorphism, the naturality square: S n S m B n B m is a pullback. The pushout B m ⊔ A m A n is a van Kampen colimit because the map A m → B m is acomplemented inclusion, it hence follows that we have a pullback square: Constructively, for part (ii) one needs to check also that the relevant objects are cofibrant in sSet . The simplicesand their boundaries are cofibrant in sSet by [GSS19, Lemma 1.3.5] and the horns by [GSS19, Lemma 1.4.9]. n ⊔ S ′ n S ′ m S m B n ⊔ A n A m B m and hence as the bottom map is a complemented inclusion by assumption, the top map is also acomplemented inclusion. This shows that S ′ → S is a cofibration.As discussed just before Lemma 1.8, the slice s E ↓ X is enriched over simplicial sets and hascotensors by finite simplicial sets. Under the present hypotheses, it also has tensors by finite (andeven countable) simplicial sets, which are simply tensors in the underlying category s E . Proposition 5.8.
Let X ∈ s E . (i) Pushout products of cofibrations in s E ↓ X exist. Moreover, cofibrations in s E ↓ X are closedunder pushout product. (ii) The pushout tensor properties of Proposition 5.5 hold in in s E ↓ X . (iii) The pullback cotensor properties of part (i) of Lemma 1.5 hold also in s E ↓ X .Proof. For part (i), recall that pushout products in s E ↓ X are computed from pushout productsin s E by pulling back along the diagonal X → X × X . Since the latter is a monomorphism,the conclusion follows from Proposition 5.3 and Lemma 5.7. For part (ii), note that the forgetfulfunctor s E ↓ X → s E preserves tensors and pushouts and thus the pushout tensor properties followdirectly from Proposition 5.5. Part (iii) was already established as Lemma 1.8, but now it alsofollows by the tensor-cotensor adjunction. Proposition 5.9. (i)
Let f : X → Y be a morphism in s E . If X is cofibrant, then the pullback functor f ∗ : s E ↓ Y → s E ↓ X preserves cofibrations. (ii) Let A → X and B → X be morphisms in s E . If A and B are cofibrant, then so is A × X B . (iii) Cofibrant objects in s E are closed under finite limits.Proof. For (i), if A → B is a cofibration over Y , then its pullback along f : X → Y coincideswith the pushout product of A → B and ∅ → X in s E ↓ Y , which is a cofibration by part (i) ofProposition 5.3. Part (ii) is a special case of part (i). Finally, for part (iii), it suffices to check thatcofibrant objects are closed under pullback and that the terminal object is cofibrant. The formerfollows from part (ii). The latter follows by definition since 0 → Pushforward along cofibrations
This section and Sections 7, 8 and 9 constitute the third part of the paper, in which we showhow the two weak factorisation systems of Section 4 give rise to the effective model structure (The-orem 9.9). For this, we shall work with a fixed countably lextensive category E . We do not assumethat the category E is (locally) Cartesian closed, but we establish the existence of certain exponen-tials and pushforwards required by our argument. We also provide a criterion for the cofibrancy ofsome of these constructions. We begin with a few remarks on exponentiable maps. Proposition 6.1.
Let f : X → Y in E . Then, the following are equivalent: (i) the pullback functor f ∗ : E ↓ Y → E ↓ X has a right adjoint f ∗ : E ↓ X → E ↓ Y , (ii) X is exponentiable as an object of E ↓ Y .Proof. This follows from [Joh02, Lemma A1.5.2 (i)] and (the proof of) [Joh02, Corollary A1.5.3].When the equivalent conditions of Proposition 6.1 hold, we say that f is exponentiable and referto the right adjoint f ∗ as the pushforward along f . (It is also known as the dependent product along f .) Example 6.2.
Let S be a finite set. Then, S ∈ E defined in (2.2) is exponentiable in E andthe exponential of X by S is the product X S . Indeed, as finite coproducts in E are universal, S × X ∼ = ` s ∈ S X . Hence, a map S × A → X is the same as an S -indexed collection of maps A → X , that is the same as a map A → X S . Proposition 6.3.
Let
W XY Z vug f be a pullback square in E . If f is exponentiable, then so is g and the canonical natural transforma-tion u ∗ f ∗ → g ∗ v ∗ is an isomorphism.Proof. This follows from [Joh02, Lemma A1.5.2 (ii)] applied in the slice category over Z . If K isan object over W , the pushforward g ∗ K is constructed explicitly as the pullback: g ∗ K f ∗ KY f ∗ W where the bottom arrow is the unit of adjunction Y → f ∗ f ∗ Y = f ∗ W .32 roposition 6.4. Let D be a small category and f • : X • → Y • a natural transformation betweentwo D -diagrams in E such that f • is Cartesian, f d is exponentiable for every d ∈ D , and Y • has avan Kampen colimit in E . Then the colimit map f : colim d ∈ D X d → colim d ∈ D Y d is exponentiable, and up to the equivalences E ↓ colim D X d ≃ lim D ( E ↓ X d ) , E ↓ colim D Y d ≃ lim D ( E ↓ Y d ) , the functor f ∗ coincides with the collection of functors ( f d ) ∗ .Proof. The claim follows from a general fact. If F : A → B is a pseudo-natural transformationbetween two diagrams A , B : D → Cat of categories such that each F d has a right adjoint R d andfor each naturality square of F d the Beck–Chevalley conditions are satisfied, then the isomorphismsgiven by the Beck–Chevalley condition exhibit R d : B d → A d as a pseudo-natural transformation,and lim R d is a right adjoint to lim F d , with the unit and counit of this adjunction being levelwisethe unit and counit of the adjunction F d ⊣ R d .We now move on to discuss how exponentiability interacts with cofibrancy. In particular, theaim of the rest of the section is to prove the following result. Theorem 6.5.
Let i : A → B be a cofibration between cofibrant object in s E . Then, (i) i is exponentiable, (ii) i ∗ sends cofibrant objects to cofibrant objects. We will prove this theorem by a saturation argument. For this purpose, we introduce now theclass G of cofibrations between cofibrant objects that satisfy the conclusion of the theorem.Assume i : A → B an exponentiable monomorphism in E . Then, for any X ∈ E ↓ A , the unit ofthe adjunction i ∗ ⊣ i ∗ induces a pullback square X i ∗ XA B . i (6.1)Indeed, since i is a monomorphism, the counit i ∗ i ! → id of the adjunction i ! ⊣ i ∗ is invertible, andtherefore so is the unit id → i ∗ i ∗ . Lemma 6.6.
Let i : A → B be a map in G . For cofibrant X ∈ E ↓ A , the map X → i ∗ X is acofibration.Proof. The claim follows from part (i) of Proposition 5.9, since the map X → i ∗ X is a pullback ofa cofibration between cofibrant objects by (6.1) above. Proposition 6.7.
The class G is closed under pushouts along maps with cofibrant target. roof. If i : A → B is in G and f : A → X is an arbitrary arrow in s E with X cofibrant, we considerthe diagram X A AX A B . if i Then the two squares are pullbacks (because i is a monomorphism for the one on the right) thevertical maps are all exponentiable by assumption, so by Proposition 6.4, the map between thecolimit of the first row to the colimit of the second row, that is the map j : X → X ⊔ A B ,is indeed exponentiable. Moreover, still by Proposition 6.4, if K is a cofibrant object over X , itcorresponds with respect to the van Kampen pushout of the first row to the Cartesian naturaltransformation K f ∗ K f ∗ KX A A . f Hence its image by j ∗ corresponds to the Cartesian natural transformation K f ∗ K i ∗ f ∗ KX A B . f i So, by gluing along the bottom van Kampen colimit, we have a pushout square f ∗ K i ∗ f ∗ KK j ∗ K where the top arrow is a cofibration by Lemma 6.6 and the assumption that i ∈ G applied to thecofibrant object f ∗ K . It follows that j ∗ K is cofibrant. Proposition 6.8.
The class G is closed under sequential composition.Proof. The class G is clearly closed under finite composition. Given an ω -chain A i A i A i . . . of arrows in G , we consider the diagram: 34 A A . . .A A A . . . .Each vertical map is in G as a composite of maps in G ; each square is a pullback as all thesemaps are monomorphisms, so by Proposition 6.4, the comparison map j : A → colim A i betweenthe two colimit is exponentiable. If K is a cofibrant object over A , then again by Proposition 6.4its image by j ∗ corresponds to the Cartesian natural transformation: K K K . . .A A A . . . where K = K and K n +1 = ( i n ) ∗ K n , hence all the maps in the top row are cofibrations, and so j ∗ K = colim K i is cofibrant. Proposition 6.9.
The class G is closed under tensors by objects of E .Proof. Let i : A B an arrow in G , and let X an object of E . The square A × X B × XA B ji is a pullback, so j is exponentiable by Proposition 6.3. Moreover, the formula for j ∗ given in theproof of Proposition 6.3 gives that K over A × X we have a pullback square j ∗ K i ∗ KB × X i ∗ ( A × X )Since i ∈ G and B × X is cofibrant, i ∗ K is cofibrant, and so j ∗ K is cofibrant, as required.In order to conclude the proof of Theorem 6.5, it remains to show that the generating cofibrations i : ∂ ∆[ n ] ∆[ n ] are in G . This is based on an explicit description of i ∗ using the characterisationof s E ↓ ∂ ∆[ n ] and s E ↓ ∆[ n ] of Lemma 2.5. Proposition 6.10.
The generating cofibrations i : ∂ ∆[ n ] ∆[ n ] are in G . roof. Under the equivalence of Lemma 2.5, the pullback functor i ∗ : s E ↓ ∆[ n ] → s E ↓ ∂ ∆[ n ] coincideswith the functor s E ∆ op ↓ ∆[ n ] → s E ∆ op ↓ ∂ ∆[ n ] obtained by reindexing along the sieve inclusion: ∆ op ↓ ∂ ∆[ n ] → ∆ op ↓ ∆[ n ], hence its right adjoint,if it exists, is the right Kan extension along this sieve inclusion. So if we prove that the pointwiseright Kan extension along this sieve inclusion exists, it will coincide with i ∗ . If F ∈ s E ↓ ∂ ∆[ n ],then this pointwise right Kan extension evaluated at ∆[ k ] → ∆[ n ] ∈ ∆ ↓ ∆[ n ] is given by the limit( i ∗ F )([ k ]) = lim p ∈ P F ( p ), where P = ∆[ a ] ∆[ k ]∆[ n ], p p not surjective .This is a limit over an infinite category so it is not guaranteed to exists, but the category P has afinite reflective category given by the objects such that the map ∆[ a ] → ∆[ k ] is injective, with thereflection given by the image factorisation of this map, and hence this limit coincides with( i ∗ F )([ k ]) = lim p ∈ P + F ( p ), where P + = ∆[ a ] ∆[ k ]∆[ n ], p p not surjective ,which is a finite limit, hence exists, which proves the existence of i ∗ .Next, we assume that F is cofibrant, and we will show that i ∗ F is cofibrant. That is, given adegeneracy [ k ] _ [ k ′ ] the action i ∗ F ([ k ′ ]) → i ∗ F ([ k ]) is a complemented inclusion (by Theorem 4.6).The map i ∗ F ([ k ]) → ∆[ n ]([ k ]) gives a decomposition of the map above into a coproduct indexedby all the map α : [ k ] → [ n ], so it is enough to show that the fiber above each such map is acomplemented inclusion. The fiber over such a map α of i ∗ F ([ k ]), is by definition of i ∗ the objectclassifying maps P → F over ∂ ∆[ n ] where P is the pullback square P ∆[ k ] ∂ ∆[ n ] ∆[ n ]. α The fiber of i ∗ F ([ k ′ ]) over α is described similarly with P ′ the pullback of ∆[ k ′ ] → ∆[ n ], and themap we are interested in is induced by the map P ′ → P obtained as the pullback of ∆[ k ′ ] → ∆[ k ].But it follows from [Hen19, Proposition 3.1.11] that a pullback of a degeneracy map is an iteratedpushout of degeneracy maps, in this case a finite such iterated pushout as P ′ is finite. As F iscofibrant, this decomposes F ( P ) → F ( P ′ ) as a composite of complemented inclusions, and henceconcludes the proof. Proof of Theorem 6.5.
We show that all cofibrations with cofibrant domain are in G . By Lemma 4.4,it suffices to show that the generating cofibrations are in G and that G is closed under operationsappearing in a cell complex. The case of generators is Proposition 6.10. Closure under tensoringby objects of E is Proposition 6.9, closure under pushout (along maps with cofibrant target) isProposition 6.7, and closure under sequential composition is Proposition 6.8.36n analysis of the proof of Theorem 6.5 shows that the assumption that A is cofibrant is notneeded for the exponentiability of i , as it is only used for the part of the argument regardingpreservation of cofibrant objects by i ∗ . We adapt the notion of a strong homotopy equivalence and the associated concepts from [GS17,Section 3] to our setting. Recall that a map f : A → B is a 0-oriented (respectively, )homotopy equivalence if there is a map g : B → A with homotopies u : gf ∼ id A and v : f g ∼ id B (respectively, u : id A ∼ gf and v : id B ∼ f g ). Such a homotopy equivalence is called strong if thehomotopies satisfy the coherence condition f u = vf .We recall the abstract characterisation of strong homotopy equivalences. The commuting square ∅ { }{ } ∆[1] !! λ λ induces maps θ : ! → λ and θ : ! → λ in the arrow category of sSet . (We will use λ ik to denote thehorn inclusion Λ i [ k ] → ∆[ k ].) Note that ! is the unit of the pushout tensor and pullback cotensor ofthe enrichment of s E in sSet . Recall that pushout tensors with levelwise complemented inclusionsbetween finite simplicial sets such as ! , λ , λ exist by Proposition 5.5. Lemma 7.1.
Let f : X → Y be a map in s E . For k ∈ { , } , the following are equivalent: (i) f is a k -oriented strong homotopy equivalence, (ii) θ k b · f : f → λ k b · f is a split monomorphism, (iii) θ k b ⋔ f : λ k b ⋔ f → f is a split epimorphism.Proof. Identical to [GS17, Lemma 4.3] and [GSS19, Lemma 3.1.1].
Corollary 7.2.
Let i be a levelwise complemented inclusion between finite simplicial sets that is astrong homotopy equivalence. For any map f in s E , the pushout tensor i b · f is a strong homotopyequivalence in s E .Proof. This is a formal consequence of the characterisation (ii) of strong homotopy equivalencesgiven by Lemma 7.1. We have θ k b · ( i b · f ) ∼ = ( θ k b × i ) b · f , a formal consequence of the isomorphism A · ( B · X ) ∼ = ( A × B ) · X natural in A, B ∈ sSet and X ∈ s E . By assumption, θ k b × i has a retraction,hence also its image under ( − ) b · f .Strong homotopy equivalences can be used to relate cofibrations and trivial cofibrations. Corollary 7.3. (i)
For a horn inclusion j ∈ J sSet and E ∈ E , the map j · E is a strong homotopy equivalence andcofibration between cofibrant objects. Any cofibration that is a strong homotopy equivalence is a trivial cofibration.Proof.
For part (i), recall from [GZ67, Chapter IV, Section 2, Paragraph 2.1.3] that the horninclusion j in sSet is a strong homotopy equivalence. By Corollary 7.2, it follows that j · E is astrong homotopy equivalence. The object E ∈ s E is cofibrant by part (i) of Proposition 5.9. ByProposition 5.5, it follows that j · E is a cofibration between cofibrant objects.Part (ii) follows from the characterisation of strong homotopy equivalences in condition (ii) ofLemma 7.1, closure of trivial cofibrations under retracts (Lemma 3.9), and Proposition 5.5 (usingthat λ and λ are trivial cofibrations). Lemma 7.4.
Let
B AX Y g f be a pullback square with X cofibrant. If, f is a k -oriented strong homotopy equivalence, where k ∈ { , } , then so is g .Proof. This is identical to [GSS19, Lemma 3.1.3], but played out in s E cof instead of sSet cof . Thepushout product with { } → ∆[1] (for k = 0) becomes a pushout tensor, which sends the cofibration ∅ → X to a trivial cofibration by Proposition 5.5. Corollary 7.5.
Let f : X ։ Y be a Kan fibration with X cofibrant. The pullback functor f ∗ : E ↓ Y → E ↓ X preserves maps that in s E are strong homotopy equivalences with cofibrant target.Proof. This follows from Lemma 7.4 using part (ii) of Lemma 1.5 and stability of cofibrant objectsunder pullback along maps with cofibrant source using part (ii) of Proposition 5.9.
Proposition 7.6 (Frobenius property) . Let f : X ։ Y be a Kan fibration with X cofibrant. Thepullback functor f ∗ : E ↓ Y → E ↓ X preserves trivial cofibrations.Proof. Let j be a trivial cofibration over Y . By Proposition 3.17, its underlying map in s E can bewritten as a retract of a J s E -cell complex j ′ . The retraction (including j ′ ) lifts uniquely to the sliceover Y . Since functors preserve retracts, this makes f ∗ j a retract of f ∗ j ′ . By Lemma 3.9, it willthus suffice to show that f ∗ j ′ is a trivial cofibration.Recall that J s E consists of levelwise complemented inclusions. By countable lextensivity, Lemma 3.8,and Corollary 2.12, the pullback functor f ∗ preserves the colimits (countable coproducts, pushouts,sequential colimit) forming the cell complex j ′ . By Lemma 3.9, it thus remains to show that f ∗ sends to a trivial cofibration any map that in s E is of the form E × j ′′ where E ∈ s E and j ′′ ∈ J sSet .Using Lemma 5.4, this simplifies to j ′′ · E . Here, we see E as a constant simplicial object in E .By part (i) of Corollary 7.3, j ′′ · E is a strong homotopy equivalence and cofibration betweencofibrant objects. By Corollary 7.5, f ∗ ( j ′′ · E ) is a strong homotopy equivalence (using that f is aKan fibration). By part (i), f ∗ ( j ′′ · E ) is a cofibration between cofibrant objects. By part (ii) ofCorollary 7.3, we conclude that f ∗ ( j ′′ · E ) is a trivial cofibration.38 Fibration extension properties
In this section, we establish two important ingredients in the construction of the effective modelstructure: the trivial fibration extension property (Proposition 8.5) and the fibration extensionproperty (Proposition 8.13). These arguments are based on the equivalence extension property(Proposition 8.3). We work purely within the cofibrant fragment s E cof of s E . Our earlier prelimi-naries allow us to prove the equivalence extension property in s E cof following [Sat17, Proposition 5.1]and [GSS19, Proposition 3.2.1].We begin with some observations on homotopy equivalences, which we introduced in Section 1,and an analysis of the restriction of the fibration category structure on s E ↡ X established inTheorem 1.9 to cofibrant objects. Since the tensor of X ∈ s E with a finite simplicial set exists andis defined by the formula in (2.1), we may equivalently write a homotopy H between f , f : X → Y in s E or one of its slices, which was defined using cotensors in (1.3), via a map H : ∆[1] · X → Y . (8.1)In E and its slices, the homotopy relation between maps with cofibrant source and fibrant target isan equivalence relation. This is a formal consequence of part (i) of Lemma 1.5 and Lemma 1.8. Itfollows that homotopy equivalences between cofibrant and fibrant objects compose as usual. Proposition 8.1. (i)
For every X ∈ s E , trivial cofibrations in s E ↡ X are homotopy equivalences. (ii) Trivial fibrations X → Y in s E cof are homotopy equivalences over Y .Proof. For part (i), in E ↡ X , given a trivial cofibration A → B , we take a lift A A × B (∆[1] ⋔ B ) B B .Here, the right map is a composition of the pullback cotensor with ∂ ∆[1] → ∆[1] of B → A →
1, hence a fibration by parts (i) and (ii) of Lemma 1.5. The lift exhibits A → B as a strong deformation retract, in particular a homotopy equivalence.For part (ii), given a fibration X → Y in E cof , we take a lift X XY ⊔ X (∆[1] · X ) Y .Here, the left map is a composition of a pushout of ∅ → Y and the pushout tensor with ∂ ∆[1] → ∆[1] of ∅ → X , hence a cofibration by Lemma 3.9 and Proposition 5.5. The lift exhibits X → Y as the dual of a strong deformation retract, in particular a homotopy equivalence over Y .39 roposition 8.2. Let X ∈ s E cof . The fibration category structure on s E ↡ X of Theorem 1.9restricts to s E cof ↡ X . Path objects are given by cotensor with ∆[1] . The weak equivalences coincidewith homotopy equivalences over X .Proof. By part (iii) of Proposition 5.9, s E cof ↡ X has finite limits and they are computed as in s E ↓ X .By part (iii) of Lemma 5.6, cotensor with ∆[1] over X preserves cofibrant objects. Thus, all aspectsof the fibration category s E ↡ X of Theorem 1.9 restrict to cofibrant objects. This includes pathobjects, which are given by cotensor with ∆[1].It remains to show that pointwise weak equivalences in s E cof ↡ X coincide with homotopy equiv-alences over X . Every homotopy equivalence is a pointwise weak equivalence by Proposition 1.10.For the reverse direction, we use the mapping path space factorisation in s E cof ↡ X , which has ahomotopy equivalence over X as first factor and fibration as second factor. Since pointwise weakequivalences and homotopy equivalences over X satisfy the 2-out-of-3 property, it suffices to showthat every pointwise weak equivalence that is a fibration (hence a trivial fibration) is a homotopyequivalence over X . This is part (ii) of Proposition 8.1. Proposition 8.3 (Equivalence extension property) . In s E cof , consider the solid part of the diagram X Y X Y A B ∼ ∼ i (8.2) where the lower square is a pullback and X → X is a homotopy equivalence over A . Then thereis Y as indicated such that the back square is a pullback and Y → Y is a homotopy equivalenceover B .Proof. The proof of [GSS19, Proposition 3.2.1] applies, but played out in s E cof instead of sSet cof .We limit ourselves to listing the key claims used in the proof and why they hold in our setting. • The slice categories s E cof ↡ A and s E cof ↡ B admit fibration category structures, established inProposition 8.2, in which weak equivalences are given by fiberwise homotopy equivalences. • The dependent product functor i ∗ along i exists and preserves cofibrant objects, as shown inTheorem 6.5. • The functor i ∗ preserves trivial fibrations, which follows by adjointness since i ∗ preservescofibrations, as stated in part (i) of Proposition 5.9. • In the slice over B , pullback cotensor with a cofibration preserves trivial fibrations, whichholds by Lemma 1.8.In s E cof , we say that a (trivial) fibration X ։ A extends along a map A → B if there is apullback square X YA B (8.3)40ith the extension Y → B of X → A again a (trivial) fibration. If A → B has this property for all(trivial) fibrations X ։ A , we say that it has the (trivial) fibration extension property . Lemma 8.4.
Let f and g be composable maps in s E cof . If g ◦ f has the (trivial) fibration extensionproperty, then so does f .Proof. We extend along f by extending along g ◦ f and pulling back along g (using part (ii) ofLemma 1.5 and part (ii) of Proposition 5.9). Proposition 8.5 (Trivial fibration extension property) . Cofibrations in s E have the trivial fibrationextension property.Proof. This is the special case of Proposition 8.3 where X → A and Y → B are the identities on A and B , respectively. We use Theorem 1.9 and Proposition 4.1 to go between trivial fibrationsand fibrations that are weak equivalences. Lemma 8.6.
Let p : X ։ ∆[1] · A be fibration in s E with A and X cofibrant. Then there is ahomotopy equivalence between X | { }· A and X | { }· A over A .Proof. Take the pullback P ∆[1] ⋔ XA ∆[1] ⋔ (∆[1] · A ). ∆[1] ⋔ p Here, the bottom map is the unit of the tensor-cotensor adjunction. The right map is a fibrationby part (i) of Lemma 1.5, hence the left map is a fibration by part (ii) of Lemma 1.5. The top rightobject is cofibrant is cofibrant by part (i) of Lemma 1.5 and part (ii), hence the top left object iscofibrant by part (ii).We will argue that there are trivial fibrations from P to X | { }· A and X | { }· A over A . Thesetrivial fibrations are homotopy equivalences over A by part (ii) of Proposition 8.1. Inverting andcomposing them as needed gives the desired weak equivalence.We only construct the trivial fibration from P to X | { }· A (the other case is dual). Consider thediagram P ∆[1] ⋔ XX | { }· A X × ∆[1] · A ∆[1] ⋔ (∆[1] · A ) XA ∆[1] ⋔ (∆[1] · A ) ∆[1] · A . λ ⋔ (∆[1] · A ) λ b ⋔ p p The two composite squares and the bottom right square are pullbacks by construction. Pullbackpasting induces the top left map and makes the top left square a pullback. The top middle map is atrivial fibration by part (i) of Lemma 1.5, hence so is the top left map by part (ii) of Lemma 1.5.Our aim now is to prove the fibration extension property for trivial cofibrations in s E cof . For thispurpose, we introduce the class G of cofibrations in s E cof that have the fibration extension property.41 emma 8.7. The class G contains cofibrations in s E cof that are strong homotopy equivalences.Proof. Let A → B be a cofibration in s E cof and 0-oriented strong homotopy equivalence (the 1-oriented case is dual). We will solve the extension problem (8.3). By the characterisation of stronghomotopy equivalences given by part (3) of Lemma 7.1, we have a retract diagram A (∆[1] · A ) ⊔ { }· A ( { } · B ) AB ∆[1] · B B . λ · B (8.4)Let Z → ∆[1] · A ⊔ { }· A { } · B denote the pullback of X → A along the top right map. Pullingback Z to ∆[1] · A , { } · A and { } · B (the components of its base pushout), we obtain the solidpart of the diagram Z | { }· A YZ | { }· A Z | { }· B A B , ∼ ∼ with lower square a pullback. Here, the weak equivalences over A is given by Lemma 8.6. We thencomplete the diagram using Proposition 8.3, making the back square a pullback. Note that Z | { }· A is isomorphic to X over A by the retract (8.4). The extension in (8.3) is then given by Y ։ B . Corollary 8.8.
For a horn inclusion j ∈ J sSet and E ∈ E , we have j · E ∈ G .Proof. This is the application of Lemma 8.7 to part (i) of Corollary 7.3.
Lemma 8.9.
The class G is closed under countable coproducts.Proof. Let A i → B i be a family of maps in G for i ∈ I countable. Note that ` i ∈ I A i → ` i ∈ I B i is acofibration between cofibrant objects by Lemma 3.9. Suppose we are given a fibration X → ` i ∈ I A i in s E cof . We aim to extend it along ` i ∈ I A i → ` i ∈ I B i . Note that ` i ∈ I B i is a van Kampen colimitsince s E is countably lextensive.For each i ∈ I , we pull it back to a fibration X i → A i (with X i cofibrant by part (ii)) andextend it to a fibration Y i → B i . We take their coproduct ` i ∈ I Y i → ` i ∈ I B i . This is a fibrationby part (i) of Lemma 3.18. Its domain is cofibrant by Lemma 3.9. By effectivity, it pulls backalong A i → ` i ∈ I B i to the map X i → A i for i ∈ I . By universality, it thus pulls back along ` i ∈ I A i → ` i ∈ I B i to the original fibration X → ` i ∈ I A i . Lemma 8.10.
The class G is closed under pushouts in s E along maps with cofibrant target.Proof. Consider a pushout square
A A ′ B B ′ . ∈ G A ′ cofibrant. Note that A ′ → B ′ is a cofibration between cofibrant objects by Lemma 3.9. Thepushout is van Kampen by part (i) of Corollary 2.12. Suppose we are given a fibration X ′ ։ A ′ in s E cof . We aim to extend it along A ′ → B ′ .We pull the given fibration back along A → A ′ to a fibration X ։ A (here, X is cofibrant bypart (ii)) and extend it to a fibration Y ։ B . Let Y ′ → B ′ be the pushout in the arrow category ofthese three maps. By effectivity, it pulls back to them. It is a fibration by part (ii) of Lemma 3.18.By part (i), X → Y is a cofibration, hence so is X ′ → Y ′ by Lemma 3.9. This makes Y ′ cofibrant.We check that Y ′ → B ′ is a fibration using Proposition 3.4. For each horn inclusion j ∈ J sSet ,we construct a section of b ev j ( Y ′ → B ′ ) given sections of b ev j ( X ′ → A ′ ) and b ev j ( Y → B ). We pullthe section of b ev j ( X ′ → A ′ ) back to a section of b ev j ( X → A ) and then extend it using Lemma 3.13to a section of b ev j ( Y → B ). The goal follows by Lemma 2.15 and functoriality of colimits. Lemma 8.11.
The class G is closed under sequential colimits.Proof. Consider the colimit B of a sequential diagram A A . . . . ∈ G ∈ G Note that it is van Kampen by part (ii) of Corollary 2.12. Suppose we are given a fibration X ։ A in s E cof . We aim to extend it along A → B .By induction on k , we extend to a fibration X k ։ A k . The maps X k → X k +1 are cofibrationsby part (i). In the end, we take the colimit and obtain a map Y → B . By effectivity, it pulls backto the maps X k ։ A k . It is a fibration by part (iii) of Lemma 3.18. Note that Y is cofibrant byLemma 3.9. Lemma 8.12.
The class G is closed under codomain retracts.Proof. This is an instance of Lemma 8.4.
Proposition 8.13 (Fibration extension property) . Trivial cofibrations in s E cof have the fibrationextension property.Proof. We have to show that G includes all trivial cofibrations between cofibrant objects. ByProposition 3.17, any such trivial cofibration can be written as a codomain retract of a sequentialcolimit of pushouts of countable coproducts of tensors with objects of E of maps in J s E . Byinduction, all the stages of the sequential colimit are cofibrant. This means that the above pushoutsquares all consist of cofibrant objects. The claim now follows starting from Corollary 8.8 using theclosure properties of G given by Lemmas 8.9, 8.10, 8.11 and 8.12. The main goal of this section is to establish the existence of the effective model structure. Sincethe categories with which we work have finite limits but do not necessarily have finite colimits, itis appropriate to consider a slight generalisation of the usual notion of a model structure. For acategory E with an initial object and a terminal object, a model structure on E consists of threeclasses of maps W , C , F such that • ( C , F ∩ W ) and ( C ∩ W , F ) are weak factorisation systems;43 W satisfies the 2-out-of-3 property; • E has pushouts along maps in C ; • E has pullbacks along maps in F .It can then be shown that W is closed under retracts, as the known proof of this fact (see [JT07,Proposition 7.8] and [Rie14, Lemma 11.3.3]) applies also assuming only the restricted limits andcolimits above. Thus, when E is finitely complete and cocomplete, this notion is equivalent to theusual one. Similarly, a model structure is determined by two of its three classes of maps also in thissetting.Let us now fix a countably lextensive category E . The existence of the effective model structureon s E will be a formal consequence of the Frobenius property of Section 7, the (trivial) fibrationextension property of Section 8, and elementary properties of the two weak factorisation systems ofTheorem 4.2. To this end, we encapsulate what is used from Section 8 as a collection of extensionoperations that all follow the same pattern. Lemma 9.1.
The following hold in s E cof . (i) Let A → B be a cofibration and X → A be a trivial fibration. There is a pullback square X YA B .with X → Y a cofibration and Y → B a trivial fibration. (ii) Let A → B be a trivial cofibration and X → A be a fibration. There is a pullback square X YA B .with X → Y a trivial cofibration and Y → B a fibration. (iii) Let A → B be a trivial cofibration and X → A be a trivial fibration. There is a pullbacksquare X YA B .with X → Y a trivial cofibration and Y → B a trivial fibration.Proof. Part (i) is the combination of Proposition 8.5 with part (i) of Proposition 5.9. Part (ii)is the combination of Proposition 8.13 with Proposition 7.6. Part (iii) follows from part (i) usingProposition 7.6 (with Proposition 5.2). 44ecall from Section 1 that a map X → Y in s E fib is a weak equivalence in the fibration categoryof Theorem 1.7 if and only if it is a pointwise weak equivalence in the sense of Definition 1.6, i.e.,Hom sSet ( E, X ) → Hom sSet ( E, Y ) is a weak homotopy equivalence of simplicial sets for all E ∈ E .Restricting to cofibrant objects, we obtain a notion of weak equivalence in s E cof , fib that satisfies2-out-of-3 and interacts as expected with cofibrations and fibrations, as recollected below. Lemma 9.2. In s E cof , fib , we have: (i) a cofibration is a trivial cofibration exactly if it is a weak equivalence, (ii) a fibration is a trivial fibration exactly if it is a weak equivalence.Proof. Part (ii) is a corollary of Proposition 4.1. For part (i), the forward direction is the combi-nation of part (i) of Proposition 8.1 and Proposition 1.10. With this, the reverse direction followsby the retract argument.In the following, we fix the following terminology regarding the weak factorisation systems ofTheorem 4.2. A fibrant replacement of X ∈ s E is a trivial cofibration X → X ′ with X ′ fibrant.By a fibrant replacement of a diagram, we mean a levelwise fibrant replacement: given a diagram X : S → s E , this is a diagram X ′ : S → s E fib with a natural transformation X → X ′ that is levelwisea trivial cofibration. If S is a finite Reedy category, we can always construct such a replacementusing Theorem 3.14 and the Reedy process. In particular, for [1] seen as a direct category, weobtain a fibrant replacement of any arrow that we call canonical . Note that the canonical fibrantreplacement preserves trivial cofibrations. We use dual terminology for cofibrant replacement .Let us write W cof for the class of maps in s E cof whose canonical fibrant replacement is a weakequivalence in s E cof , fib . This will be the class of weak equivalences in the model structure on s E cof to be established in Proposition 9.6. Lemma 9.3.
Let A → B in s E cof . Then, the the following are equivalent: (i) the map A → B is in W cof , (ii) the map A → B has a fibrant replacement that is a weak equivalence in s E cof , fib , (iii) all fibrant replacements of the map A → B are weak equivalences in s E cof , fib .Proof. This is a standard argument and goes exactly as in [GSS19, Lemma 3.3.1]. What is used ispart (i) of Corollary 2.12 with the fact that trivial cofibrations are levelwise complemented inclusions(Proposition 3.17), and closure properties of trivial cofibrations (Lemma 3.9), the forward directionof part (i) of Lemma 9.2, and 2-out-of-3 for weak equivalences in s E cof , fib . Corollary 9.4.
The class W cof satisfies the 2-out-of-3 property.Proof. Using Lemma 9.3 with levelwise fibrant replacement of the given 2-out-of-3 diagram, thisreduces to closure of weak equivalences in s E cof , fib under 2-out-of-3. This is part of Theorem 1.7. Lemma 9.5. In s E cof , a fibration is a trivial fibration if and only if it is in W cof . roof. Let X → Y be a fibration in s E cof . Take a fibrant replacement Y → Y .If X → Y is a trivial fibration, we extend it to a trivial fibration X → Y using part (iii) ofLemma 9.1. Then X → Y is a weak equivalence by part (ii) of Lemma 9.2, hence X → Y is in W cof by Lemma 9.3.In the reverse direction, we extend X → Y to a fibration X → Y using part (ii) of Lemma 9.1.If X → Y is in W cof , then X → Y is a weak equivalence by Lemma 9.3, hence a trivial fibration bypart (ii) of Lemma 9.2. Then its pullback X → Y is a trivial fibration by part (ii) of Lemma 1.5. Proposition 9.6.
The category s E cof admits a model structure with weak equivalences W cof andthe two weak factorisation systems of Theorem 4.2.Proof. First note that s E cof has finite limits by part (iii) of Proposition 5.9, an initial object by lex-tensivity, and pushouts of cofibrations by part (i) of Corollary 2.12 (since cofibrations are levelwisecomplemented inclusions by Proposition 3.17). The class W cof satisfies 2-out-of-3 by Corollary 9.4.It remains to show that a (co)fibration is trivial exactly if it is a weak equivalence. For fibrations,this is Lemma 9.5. For cofibrations, the forward direction is immediate using Lemma 9.7: a giventrivial cofibration has as fibrant replacement the identity on a fibrant replacement of its codomain;but identities are weak equivalences in s E cof , fib by Theorem 1.7. The backward direction followsfrom this by the retract argument.We write W for the class of maps in s E whose canonical cofibrant replacement is in W cof .This will be the class of weak equivalences of the effective model structure, to be established inTheorem 9.9. Lemma 9.7.
Let A → B in s E . Then, the following are equivalent: (i) the map A → B is in W , (ii) the map A → B has a cofibrant replacement in W cof , (iii) all cofibrant replacements of the map A → B are in W cof .Proof. This is a standard argument, dual to the one of Lemma 9.3. What is used is closure propertiesof trivial fibrations (part (ii) of Lemma 1.5) and the model structure on E cof of Proposition 9.6. Corollary 9.8.
The class W satisfies the 2-out-of-3 property.Proof. This is analogous to the proof of Corollary 9.4.We can finally establish the existence of the effective model structure on s E . Theorem 9.9 (The effective model structure) . Let E be a countably lextensive category. (i) The category s E of simplicial objects in E admits a model structure determined by the twoweak factorisation systems of Theorem 4.2. (ii) A map between fibrant objects is a weak equivalence in this model structure if and only if it isa pointwise weak equivalence in the sense of Definition 1.6. (iii)
More generally, for X ∈ s E , a map in s E ↡ X is a weak equivalence exactly if and only if it isa pointwise weak equivalence in s E in the sense of Definition 1.6. roof. First note that s E has finite limits by lextensivity and the required colimits of a modelstructure by the same reasoning used for Proposition 9.6. We define the class of weak equivalencesto be W . It satisfies 2-out-of-3 by Corollary 9.4. It remains to show that a (co)fibration is trivialexactly if it is a weak equivalence.Due do our definition of W , we get for free that every trivial fibration is a weak equivalence,dually to the reasoning for trivial cofibrations in Proposition 9.6.For the reverse direction, let X → Y be a fibration and weak equivalence. Let b X → b Y denoteits canonical cofibrant replacement. This is the Reedy cofibrant replacement over the inverse cat-egory [1], hence again a fibration. Since b X → b Y is a fibration and weak equivalence in E cof , it isa trivial fibration by Proposition 9.6. The composite b X → Y is a trivial fibration by part (ii) ofLemma 1.5. By part (iii) of Lemma 1.5, we deduce that X → Y is a trivial fibration.Let A → B be a trivial cofibration. Take a cofibrant replacement b B → B . Let b A → A be its pullback along A → B . Then b A is cofibrant by Lemma 5.7 since trivial cofibrations aremonomorphisms by Proposition 5.2, b A → A is a trivial fibration by part (ii) of Lemma 1.5, and b A → b B is a trivial cofibration by Proposition 7.6. In particular, b A → b B is a cofibrant replacementof A → B . Since it is a trivial cofibration, it is a weak equivalence in E cof by Proposition 9.6. ByLemma 9.7, this makes A → B is a weak equivalence.It remains to show that every cofibration that is a weak equivalence is a trivial cofibration. Asin Proposition 9.6, this follows from what we have already established by the retract argument.This finishes the verification of part (i). Parts (ii) and (iii) follow since every model structureinduces a fibration category structure on its fibrant objects (and those of its slices) and the weakequivalences in a fibration category are determined by its fibrations and trivial fibrations. In ourcase, we obtain the fibration categories of Theorems 1.7 and 1.9.By part (ii) of Theorem 9.9, a map is a weak equivalence in the effective model structure if andonly if its fibrant replacement is a pointwise weak equivalence. This gives us a description of weakequivalences independent from the class W used in the construction of the model structure. Remark 9.10.
When E , and hence s E , is a locally presentable, then one can use the enrichedsmall object argument of [Rie14, Chapter 13] to produce the two weak factorisation systems on s E whose fibrations and trivial fibrations are as in Definition 1.3. Theorem 1.7 then implies that s E isa weak model category, for example using the dual of [Hen18, Proposition 2.3.3]. It then followsfrom [Hen20, Theorem 3.7] that its left saturation (in the sense of [Hen20, Theorem 4.1]) is a leftsemi-model category, and from [Hen20, Theorem 3.8] that it is also is a right semi-model category.In general, this is not quite enough to conclude that it is a Quillen model category (it is what iscalled a two-sided model category in [Hen20, Section 5]), but this is already sufficient for manyapplications.When E is an additive locally presentable category, then there is Quillen model structure on s E whose fibrations and trivial fibrations are exactly as in Definition 1.3. The additional ingredientin this case is that for A ∈ E and X ∈ s E , the object Hom sSet ( A, X ) is a simplicial abelian group,hence it always a Kan complex. This shows that when E is additive, all objects of s E are Kancomplexes, hence in the discussion above it is immediate that s E is left saturated (in the senseof [Hen20]) and as it is a saturated right semi-model category where every object is fibrant it isa Quillen model category. By the Dold–Kan correspondence, the category s E is equivalent to thecategory of chain complexes in E and under this equivalence the model structure is the so calledabsolute (or Hurewicz) model structure on chain complexes (see, e.g., [CH02]).47 Having established the existence of the effective model structure on s E , we now study some ofits properties and those of its associated ∞ -category Ho ∞ ( s E ). For a general category C equippedwith a class of weak equivalences, Ho ∞ ( C ) is the ∞ -category obtained by universally inverting weakequivalences in C . There are many known models of higher categories (as well as constructions ofHo ∞ ), the most popular being quasicategories (see, e.g., [Cis20]). Our results do not depend on aparticular choice of such model.We begin by studying the behaviour of colimits, using the notion of descent, which was intro-duced in model categories by Rezk [Rez10] as a part of development of higher topos theory. Weshow that s E and hence Ho ∞ ( s E ) satisfies descent whenever s E is countably extensive. This meansthat colimits in Ho ∞ ( s E ) satisfy the higher categorical version of the van Kampen property. In thecase of pushouts, this is spelled out in Proposition 10.1 below. As in the ordinary categorical case,a colimit in an ∞ -category C satisfies descent if and only if it is preserve by the functor from C op tothe ∞ -category of ∞ -category classified by the slice cartesian fibration. This is essentially provedin section 6.1.3 of [Lur09], see for example 6.1.3.9. Proposition 10.1 (Model structure descent for pushouts) . Let E be a countably extensive categoryand let X X X X Y Y Y Y (10.1) be a cube in s E . Assume that the bottom face is a homotopy pushout and that the left and back facesare homotopy pullbacks. Then the following are equivalent: (i) The top face is a homotopy pushout, (ii) the right and front faces are homotopy pullbacks.Proof.
Let us view [1] as a Reedy category consisting only of face maps. We consider the Reedymodel structure [ D op , s E ] of s E over D = [1] × ([1] × [1]) op . Recall from the beginning of Section 9that we regard only certain (co)limits to be part of a model structure; the theory of Reedy modelstructures makes sense in this setting as seen in Section 4 for the case of the Reedy weak factorisationsystem over ∆.The given cube (10.1) forms an object of this category by sending (0 , a, b ) to Y ab and (1 , a, b )to X ab . Recall that weak equivalences in the Reedy model structure are levelwise and homotopypushouts and pullbacks are invariant under levelwise weak equivalences. We replace the given cubeby a cofibrant and fibrant object. This reduces the claim to the case of (10.1) where all object arecofibrant and fibrant, all horizontal maps are cofibrations, and all vertical maps are fibrations.Let us check the direction from (i) to (ii), i.e., universality. Take the pullback of the bottomface along X ։ Y . Since all vertical faces in (10.1) are homotopy pullbacks, we obtain a square48eakly equivalent to the top face. This reduces the claim to the situation where in addition allvertical faces in (10.1) are pullbacks. Note that the cofibrancy assumptions are preserved by part (i)of Proposition 5.9.Denote Q the pushout in the bottom face. Since Y → Y is a levelwise complemented inclusion(Proposition 3.17), P is a van Kampen pushout by Lemma 2.9, in particular stable under pullback.From universality, we obtain a pullback square P QX Y (10.2)where P is the pushout in the top face. Since X → X and Y → Y are cofibrations, thebottom and top faces are homotopy pushouts exactly if the maps P → X and Q → Y are weakequivalences, respectively. The goal thus follows from right properness applied to (10.2).Let us check the direction from (ii) to (i), i.e., effectivity. Take the pushout in the horizontalfaces. Since all horizontal maps are cofibrations and the horizontal faces are homotopy pushouts,we obtain a cube weakly equivalent to the given cube. This reduces the goal to the situation whereall horizontal faces in (10.1) are pushouts, but note that we lose fibrancy properties involving X and Y . The cube is now determined (up to isomorphism) by just the left and back faces. Weaklyequivalent left and back faces give rise to weakly equivalent cubes.Since the back face is a homotopy pullback and the vertical maps are fibrations, the map X → Y × Y X is a weak equivalence. We apply the equivalence extension property of Proposition 8.3to this situation: X X ′ Y × Y X X Y Y i ∼ ∼ We perform the same construction in the left face, obtaining X ′ . Now, the squares X ′ X X ′ Y Y Y are weakly equivalent to the left and back faces, but are pullbacks. We have thus reduced to thesituation where additionally the left and back faces of (10.1) are pullbacks.Having strictified the given homotopy pushouts and homotopy pullbacks, we proceed as follows.The maps X → X and X → X are levelwise complemented inclusions by Proposition 3.17.The bottom pushout is van Kampen by part (i) of Corollary 2.12. In particular, the right and frontfaces are pullbacks. For them to be homotopy pullbacks, it suffices for X → Y to be a fibration.This holds by part (ii) of Lemma 3.18. 49 roposition 10.2 (Model structure descent for coproducts) . Let E be an α -extensive category, X → Y a morphism in s E and S an α -small set. Given a square X s XY s Y for each s ∈ S such that the induced morphism ` s Y s → Y is a weak equivalence, the following areequivalent: (i) the square above is a homotopy pullback for each s ∈ S , (ii) the induced morphism ` s X s → X is a weak equivalence.Proof. This follows from a simpler variant of the previous argument, for α -small coproducts insteadof pushouts. This uses part (i) instead of part (ii) of Lemma 3.18.Propositions 10.1 and 10.2 have an immediate counterpart at the ∞ -categorical level. Theorem 10.3.
Let E be an α -extensive category. The ∞ -category Ho ∞ ( s E ) has all α -smallcolimits. These colimits satisfy descent.Proof. It is a well-known fact (see, e.g., [Lur09, Theorem 4.2.4.1]) that given an ordinary diagramin a model category, its (co)cone is a homotopy (co)limit if and only if it is a (co)limit in theassociated ∞ -category. It follows from Proposition 10.1 and Proposition 10.2 that pushouts and α -coproduct exist and satisfy descent in Ho ∞ ( s E ). By [Lur09, Proposition 4.4.2.6], Ho ∞ ( s E ) hasall α -small colimits. Given that a certain colimit satisfies descent if and only if it is preservedby the contravariant functor from Ho ∞ ( s E ) to the ∞ -category of ∞ -categories classified by theslice fibration, [Lur09, Proposition 4.4.2.7] shows that this implies that all α -small colimits satisfydescent.We now move on to consider right properness of the effective model structure, which will be thekey to transfer local Cartesian closure from E to Ho ∞ ( s E ). Proposition 10.4.
Let E be a countably lextensive category. The effective model structure on s E is right proper.Proof. This follows from Proposition 7.6 using the argument in [GSS19, Proposition 4.1, Secondproof].
Theorem 10.5.
Let E be a countably lextensive category. If E is locally Cartesian closed, then the ∞ -category Ho ∞ ( s E ) is locally Cartesian closed.Proof. We first observe that if E is countably lextensive and locally Cartesian closed, then s E is alsolocally Cartesian closed. Indeed, if E is countably lextensive then s E can be realised as the categoryof internal presheaves for the category object ∆ ∈ E . Such categories of internal presheaves overan internal category in a locally Cartesian closed categories are always locally Cartesian closed.Indeed, this follows from [Joh02, Theorem A4.2.1 and Proposition B2.3.16], using exactly the sameargument as in the proof of [Joh02, Corollary B2.3.17] (which deals with the similar statement50or toposes instead of locally Cartesian closed categories). Note that we are applying these resultstaking the category D therein to be the canonical self-indexing of the base category E , which satisfiesthe assumption of having E -indexed products because of [Joh02, Lemma B1.4.7, part (iii)] since E is locally Cartesian closed.A map in Ho ∞ ( s E ) can always be represented in s E by a fibration p : X → Y between fibrantobjects with X cofibrant. Because X and Y are fibrant, the ∞ -categorical slices Ho ∞ ( s E ) ↓ X and Ho ∞ ( s E ) ↓ Y can be represented using the slice model structure s E ↓ X and s E ↓ Y , and thepullback functor Ho ∞ ( s E ) ↓ Y → Ho ∞ ( s E ) ↓ Y can be represented by the right Quillen functor p ∗ : s E ↓ Y → s E ↓ X . But p ∗ is a left adjoint functor since s E is locally Cartesian closed, it preservescofibrations by part (i) of Proposition 5.9 and it preserves trivial cofibrations by the Frobeniusproperty of Proposition 7.6. Thus, p ∗ is a left Quillen functor and hence it induces a left adjointfunctor at the level of the associated ∞ -categories, as required.We conclude this section by combining our results in the case E is a Grothendieck topos. Theorem 10.6.
Let E be a Grothendieck topos. Then ∞ -category Ho ∞ ( s E ) is locally Cartesianclosed and has all small colimits, which satisfy descent. For a Grothendieck topos E , the effective model structure on s E is typically not a model topos inthe sense of Rezk [Rez10] and Ho ∞ ( s E ) is not a higher topos in the sense of Lurie [Lur09]. Indeed,as we will see in Example 11.8, if E = Set [1] , then the category of 0-truncated objects in Ho ∞ ( s E )is neither a Grothendieck topos nor an elementary topos, as it does not have a subobject classifier.The situation is reminiscent of that of Grothendieck toposes whose exact completion is neither aGrothendieck topos nor an elementary topos [Men03].
11 A generalised Elmendorf theorem
Elmendorf’s theorem [Elm83, Ste16] states that the genuine equivariant model structure on G -spaces is equivalent to the projective model structure on presheaves of spaces on the categoryof orbits of G . In this section, we show as Theorem 11.7 that, under the assumption that thecategory E is completely lextensive and locally connected (in the sense of Definition 11.1 below),then the effective model category structure on s E models the ∞ -category of small presheaves ofspaces on the full subcategory E con of connected objects in E . Note that extension of Elmendorf’stheorem beyond the case of group action already appears in the literature (cf. [Cho15, DK84, F87]).The work in [Cho15] is especially close to what we prove in the present section. Definition 11.1.
Let E be a lextensive category. • An object X ∈ E is said to be connected if it is not the initial object and whenever X = A ⊔ B then A = ∅ or B = ∅ . • A lextensive category is said to be locally connected if every object is a van Kampen coproductof connected objects.The terminology of Definition 11.1 is compatible with the notion of a locally connected Gro-thendieck topos. For example, the category of sheaves of set over a locally connected topologicalspace is locally connected. The category of presheaves over a category I is locally connected, itsconnected objects are called the “orbit” of I , i.e., the presheaves whose category of elements is51onnected, or equivalently whose colimits is a singleton. The coproduct completion of a categorywith finite limits is also a locally connected category.Let us now fix a lextensive category E . We denote by E con the full subcategory of of E ofconnected objects. It is important to note that even if E is a Grothendieck topos, this category isin general not a small category, as the next example illustrates. Example 11.2. If E = Set [1] = Fam Set , then the connected objects of E are the objects of theform X → ∗ for an arbitrary set X . In particular E con is equivalent to the category of all sets.More generally, if C is a category with finite limits, and Fam C is its coproduct completion, then( Fam C ) con = C . Lemma 11.3.
Let X be a connected object in a lextensive category. Then Hom
Set ( X, − ) commuteswith van Kampen coproducts.Proof. Given a map f : X → ` A i , then X = ` X i where X i = X × A A i , but as X is connectedall the X i except one are the initial object. As X is itself non-initial, then exactly one of the X i isnon initial and hence X = X i and the map X → ` A i factors into X → A i for a unique i .For a possibly large category D , we write Psh D for the category of small presheaves on D , thatis the category of presheaves on D that can be written as small colimits of representables. Wedenote by sPsh D the category of small simplicial presheaves, or equivalently simplicial objects in Psh D . In general, limits of small presheaves can fail to be small, but if we assume that D has α -small limits, then Psh D also has α -small limits. This is proved in [DL07] as Theorem 4.3 appliedto Example 4.1.1. Proposition 11.4.
Let D be a category with finite limits. Then sPsh D carries the projective modelstructure, in which an arrow f : X → Y if a fibration, trivial fibration or weak equivalence if andonly if for all d ∈ D , the arrow f d : X ( d ) → Y ( d ) is one.Proof. This is proved in [CD09] under the assumption that D has all limits. However, the proofapplies unchanged if we only assume that sPsh D has finite limits, as long as do require that amodel category has all limits. The existence of the corresponding weak factorisation system followsfrom the generalised small object argument with respect to locally small class of arrows exactly asexplained in [CD09]The claim of Proposition 11.4 follows also from the assumption that sPsh D has finite limits,which is a weaker condition than the existence of finite limits in D . Remark 11.5.
The ∞ -category associated to the projective model structure on sPsh D is reallythe ∞ -category of small presheaves of spaces on D , essentially by the same argument as for smallcategories. Lemma 11.6.
Given a locally connected countably lextensive category E . (i) The restricted Yoneda embedding y :
E →
Psh ( E con ) is well-defined, fully faithful and preserveslimits and all van Kampen coproducts. (ii) The restricted Yoneda embedding y : s E → sPsh ( E con ) is well-defined, fully faithful and pre-serves limits, pushouts along a cofibration, tensoring by objects of E and sSet and colimits ofsequences of cofibrations. roof. For any connected object X ∈ E con , Hom Set ( X, − ) preserves coproducts by Lemma 11.3,hence as every object Y ∈ E is a small van Kampen coproduct of connected objects, its imageunder the restricted Yoneda embedding is a small coproduct of representables, and hence is a smallpresheaf. This proves the existence and the preservation of coproducts by the Yoneda embedding.Preservation of limits is immediate. It is fully faithful on connected objects by the Yoneda lemma,and this implies that it is fully faithful in general as morphisms between van Kampen coproductsof connected objects can be explicitly described as maps between their components.The simplicial version is just the ordinary version applied levelwise in the simplicial directionso all results of part (ii) follow immediately. For the preservation of colimits we use the fact thata functor that preserves countable coproducts preserves pushouts of complemented inclusions andcolimits of sequences of complemented inclusions, and all the colimits considered in the lemma arelevelwise of this form. Theorem 11.7 (Generalised Elmendorf’s theorem) . Let E a locally connected countably lextensivecategory. (i) A map in s E is a cofibration, fibration or weak equivalence if and only if its image by therestricted Yoneda embedding is one for the projective model structure. (ii) If E is in addition completely lextensive, then the restricted Yoneda embedding induces anequivalence between the full subcategories of cofibrant objects of s E and sPsh ( E con ) . In partic-ular it induces an equivalence of the corresponding ∞ -categories.Proof. The (cofibration, trivial fibration) and (trivial cofibration, fibration) weak factorisation sys-tems on s E are cofibrantly generated in the (non-enriched) sense of [CD09] by the classes of arrows { i · E | i ∈ I sSet , E ∈ E} and { j · E | j ∈ I sSet , E ∈ E} . As every object in E is assumed to be a (vanKampen) coproduct of connected objects, one can restrict to E ∈ E con . Because of Lemma 11.6,these generators are sent exactly to the generators of the projective model structure of sPsh ( E con ).It immediately follows that an arrow in s E is a (trivial) fibration if and only if it is one in sPsh ( E con ) as these classes are characterised by the same lifting property.Moreover, also because of Lemma 11.6 the restricted Yoneda embedding preserves coproductsand pushouts of the generating cofibrations, transfinite composition of cofibrations and retracts.Thus because of how (trivial) cofibrations are constructed in s E from the small object argument,it follows that their images in sPsh ( E con ) are projective (trivial) cofibrations. Conversely, an arrowin s E which is a (trivial) cofibration in the projective model structure on sPsh ( E con ) has the liftingproperty against all (trivial) fibrations in sPsh ( E con ), but as the restricted Yoneda embedding isfully faithful and preserves (trivial) fibrations, it follows that it also has the lifting property againstall (trivial) fibrations in s E and hence is a (trivial) cofibration in s E . This proves part (i) for(trivial) cofibrations and (trivial) fibrations, the case of equivalences also follows as an arrow is anequivalence if and only if it can be factored as trivial cofibration followed by a trivial fibration.For part (ii), we just make one additional observation. If E is completely lextensive, thenany cofibrant object in sPsh ( E con ) is in the image of the Yoneda embedding. Indeed, the imageof y contains the initial object and the generating cofibrations, and is closed under pushout ofcofibrations, transfinite composition of cofibrations and retract (because it is closed under finitelimits). Therefore, it contains all cofibrant objects. So as y is fully faithful it is an equivalence ofcategories between the categories of cofibrant objects.53n short, Theorem 11.7 says that if E is completely lextensive and locally connected, the effectivemodel category structure on s E of Theorem 9.9 models the category of small presheaves of spaceson the large category E con . Note that we cannot quite say that the restricted Yoneda embedding isa Quillen equivalence because it does not admit an adjoint in general. However it follows from thetheorem that if E has all colimits, then it is a right Quillen equivalence. Note that a very generalElmendorf’s theorem was also proved in [Cho15, Theorem 3.1], which is similar to our version inmany aspects. In fact, if we assume that E is both complete and cocomplete then we can deduceour result from Chorny’s theorem. Example 11.8.
We take E to be the category Set [1] of arrows in
Set . It is completely lextensiveand locally connected, and its connected objects are the object of the form X → Y where Y isthe singleton. So the category of connected objects identifies with the category of sets, it hencefollows by Theorem 11.7 that the category Ho ∞ ( s E ) identifies with the category of small presheavesof spaces on the category of all sets. This ∞ -category satisfies descent (all its colimits are VanKampen) and is locally cartesian closed, for example by Theorem 10.5 and Theorem 10.3. But, itis not a (locally) presentable ∞ -category, so is not an ∞ -topos in the sense of [Lur09]. It is alsonot an elementary ∞ -topos for most definitions of the term, for example its full subcategory ofset-truncated objects is the category of small presheaves of sets on the category of all sets and isnot an elementary topos as it does not have a subobject classifier. This category of set-truncatedobjects is however a pretopos (in the infinitary sense of the term) and is locally cartesian closed.
12 Semisimplicial objects and left properness
In this section, we consider the category of semisimplicial objects s + E . While its homotopytheory is overall less well-behaved than its simplicial counterpart we developed so far, it is in somerespects simpler. This allows us to derive certain properties of s E that we do not seem to be able toprove otherwise. In particular, we use these results to show that the model structure on s E is leftproper (Corollary 12.18) and to establish certain universal property of the ∞ -category associatedwith s E in Section 13.Our development will be mostly parallel to the simplicial one. We will start under the assump-tion that E has finite limits and show that the category of Kan complexes in s + E carries a structureof a fibration category. If E is countably lextensive, the category s + E also carries natural notionsof cofibrations and trivial cofibrations, but these do not fit into a model structure. (They can beorganised into certain weaker structures as discussed below in Remark 12.9.) Nonetheless, we showthat they are sufficiently well-behaved for our purposes. Indeed, a particularly simple characterisa-tion of cofibrations (they coincide with levelwise complemented inclusions, see Lemma 12.3) enablescertain arguments unavailable in s E .The critical result that is that the homotopy theories of simplicial and semisimplicial objectsin s E are equivalent (Theorem 12.6). We will show that under the assumption that E is eithercountably complete (Theorem 12.8) or countably lextensive (Theorem 12.17).We begin by introducing some basic concepts. Since these are largely analogous to the simplicialcase, we only treat them briefly, mainly to fix the notation. We write ∆ + for the subcategory of∆ consisting of the face operators (i.e., the injective maps) and s + E = [∆ op+ , E ] for the categoryof semisimplicial objects in E . In particular, s + Set is the category of semisimplicial sets. Therepresentable semisimplicial sets are denoted by ∆ + [ n ]. For any finite semisimplicial set K , we54efine the evaluation functor ev K : s E → E asev K ( X ) = R [ n ] ∈ ∆ + X K n n .The category s + Set caries a non-Cartesian closed symmetric monoidal structure whose tensoris called the geometric product and denoted by ⊠ . It is uniquely determined by the propertythat ∆ + [ m ] ⊠ ∆ + [ n ] is the semisimplicial set of non-degenerate simplices in the nerve of the poset[ m ] × [ n ].The forgetful functor U : s + Set → sSet has both the left adjoint L and the right adjoint R givenby Kan extensions along the inclusion ∆ + → ∆. The forgetful functor U : s + E → s E also has theleft or the right adjoint if E is countably lextensive (or even just finitely cocomplete) or countablycomplete, respectively. These will be used in the proofs of the two variants of this section’s maintheorem announced above.The homotopy theory of semisimplicial sets is well established. Weak homotopy equivalences aredefined as semisimplicial maps that become simplicial weak homotopy equivalences upon applyingthe functor L . The category s + Set also carries classes of (trivial) fibrations and cofibrations, definedbelow. These do not form a model structure, but they satisfy certain weaker axioms. E.g., s + Set isa weak model category (and even a right semi-model category), see [Hen19, Section 5.5]. For ourpurposes, Theorem 12.2 below is sufficient.For a finite semisimplicial set K and X ∈ s + E we define the cotensor K ⋔ X ∈ s + E by letting( K ⋔ X ) n = X (∆ + [ n ] ⊠ K )and the semisimplicial hom-objectHom s + Set ( X, Y ) n = Hom Set ( X, ∆ + [ n ] ⋔ Y ).Exactly as in the simplicial case, this makes s + E into a s + Set -enriched category with respect to thegeometric product and ⋔ becomes the cotensor for this enrichment.The boundaries ∂ ∆ + [ n ] and horns Λ k + [ n ] are defined analogously to their simplicial counterparts( ∂ ∆ + [ n ] consists of non-degenerate simplices of ∂ ∆[ n ] and similarly for Λ k + [ n ]). This gives rise tothe generating sets I s + Set = { ∂ ∆ + [ n ] → ∆ + [ n ] } and J s + Set = { Λ k + [ n ] → ∆ + [ n ] } in s + Set and I s + E = { ∂ ∆ + [ n ] → ∆ + [ n ] } and J s + E = { Λ k + [ n ] → ∆ + [ n ] } in s + E .Then a morphism X → Y in s E is a fibration if the pullback evaluation X (∆ + [ n ]) → X (Λ k + [ n ]) × Y (Λ k + [ n ]) Y (∆ + [ n ])has a section for all horn inclusions Λ k + [ n ] → ∆ + [ n ] in J s + Set and a trivial fibration if X (∆ + [ n ]) → X ( ∂ ∆ + [ n ]) × Y ( ∂ ∆ + [ n ]) Y (∆ + [ n ])has a section for all boundary inclusions ∂ ∆ + [ n ] → ∆ + [ n ] in I s + Set . Similarly, cofibrations and trivialcofibrations are defined as I s + E -cofibrations and J s + E -cofibrations in the sense of Definition 3.2. Notethat fibrations and trivial fibrations defined above coincide with J s + E -fibrations and I s + E -fibrationsby the same argument as in Proposition 4.1. 55 emma 12.1. If E is countably lextensive, then s + E carries two enriched weak factorisation systemsconsisting of: • cofibrations and trivial fibrations, • trivial cofibrations and fibrations.Proof. This follows from Theorem 3.14 with the assumptions verified exactly as in the proof ofTheorem 4.2.
Theorem 12.2.
The category of fibrant semisimplicial sets with weak homotopy equivalences asdefined above (i.e., created by the free functor L : s + Set → sSet ) is a fibration category.Proof sketch. The claim can be deduced from the existence of the fibration category of fibrantsimplicial sets in [GSS19, Theorem 2.2.2]. The proof is analogous to the proof of [GSS19, Theo-rem 2.2.2] itself and depends on the following fact. If f : X → Z is a map between simplicial setsand U f factors (in semisimplicial sets) as a composite of a cofibration i : U X → B and a fibration p : B → U Z , then f factors as a composite of i ′ : X → Y and p ′ : Y → Z such that i = U i ′ and p = U p ′ . (Note that, in particular, B = U Y , i is a cofibration and p is a fibration.) This holdsby [Ste17, Theorem 2.1 and Addendum 2.2]. It will also rely the fact that U preserves and reflectsweak equivalences by [Hen19, Lemma 2.2.1].Compared to the proof of [GSS19, Theorem 2.2.2], the present argument requires only twomodifications. First, to construct a path object on a fibrant semisimplicial set K , we first applythe fact above (with X = ∅ , Y = K and Z = 1) to obtain a simplicial Kan complex A such that U A = K . Then we obtain a path object on K by applying U to a path object on A . Second, weobserve that the facts above imply that a fibration in s + Set is acyclic if and only if it is trivial (byreducing it to the same statement in sSet ). Thus acyclic fibrations are stable under pullback. Lemma 12.3.
A map f : X → Y in s + E is a cofibration if and only if for all n the map X n → Y n is a complemented inclusion. In particular, every object of s + E is cofibrant.Proof. The claim follows already from the semisimplicial version of Proposition 4.3 since latchingobjects are empty, which is simpler to prove than Proposition 4.3 due to absence of degeneracyoperators.
Corollary 12.4. If E has finite limits, then every trivial fibration in s + E admits a section.Proof. First, note that if E is countably lextensive, this follows from Lemmas 12.1 and 12.3. If E ismerely finitely complete, then Fam ω E is countably lextensive and the conclusion holds since thefunctor s + E → s + Fam ω E is fully faithful, cf. the explicit construction of Fam α in Example 2.8.A morphism X → Y between Kan complexes in s + E is a pointwise weak equivalence ifHom s + Set ( E, X ) → Hom s + Set ( E, Y )is a weak equivalence in s + Set for all E ∈ E . This is non-constructive, because of the use of [Ste17]. An alternative argument which works constructively canbe found in [Hen19, Theorem 5.5.6]. It shows that semisimplicial set have a weak model structure analogous to theKan–Quillen model structure. Given that even constructively all semisimplicial sets are cofibrant this is enough toobtain that the full subcategory of fibrant objects is a fibration category. heorem 12.5. Pointwise weak equivalences, fibrations and trivial fibrations equip the category ofKan complexes in s + E with the structure of a fibration category.Proof. The proof is entirely analogous to the proof of Theorem 1.7 except for the construction ofpath objects. A path object on X ∈ s + E can be constructed as X → ∆ + [1] ⋔ X ։ X × X as before.However, there is no semisimplicial map ∆ + [1] → ∆ + [0] (i.e., ∆ + [0] does not admit a cylinderobject) and so the morphism X → ∆ + [1] ⋔ X cannot be induced by functoriality of cotensors. Theproblem can be fixed by constructing a “weak cylinder object” on ∆ + [0] in the sense of [Hen18].There is a unique map Λ [2] → ∆ + [1]. It sends both 1-simplices to the unique 1-simplex of∆ + [1]. We define D to be the pushout of this map along the trivial cofibration Λ [2] → ∆ + [2]:Λ [2] ∆ + [1]∆ + [2] D .Thus D has two 0-simplices b and x , two 1-simplices f : b → x and e : b → b and a unique 2-simplexthat witnesses that f ◦ e ∼ e . Informally speaking, this forces e to behave as an “identity cell” of b .More precisely, we obtain a diagram ∂ ∆ + [1] ∆ + [0]∆ + [1] D be ∼ which upon cotensoring into X ∈ s + E yields X × X X ∆ + [1] ⋔ X D ⋔ X . ∼ When X is a Kan complex, the right vertical morphism is a trivial fibration and hence it has asection by Corollary 12.4. We obtain the required factorisation by composing D ⋔ X −→ ∼ ∆ + [1] ⋔ X with such section. This last map is a pointwise weak equivalence, because applying Hom s + Set ( E, − )to it gives, up to isomorphism, the map D ⋔ Hom s + Set ( E, X ) → ∆ + [1] ⋔ Hom s + Set ( E, X )which is a semisimplicial weak equivalence for each fibrant semisimplicial set Hom s + Set ( E, X ),for example because both evaluation maps to Hom s + Set ( E, X ) are trivial fibrations as the weakfactorisation systems on s + Set are compatible to the monoidal structure on s + Set (see for eg. Theo-rem 5.5.6.(iii) of [Hen19]).The following theorem is the main result of this section. It is valid under two separate setsof assumptions which require two independent proofs. Thus we will consider them separately asTheorem 12.8 and Theorem 12.17. 57 heorem 12.6. If E is either countably lextensive or countably complete, then the forgetful functor s E → s + E induces an equivalence of fibration categories between the fibration categories of Theo-rems 1.7 and 12.5. We start with the case of a category E with countable limits, this is the proof that relies on theadjunction U ⊣ R . Proposition 12.7. If E is countably complete, then the forgetful functor U : s E → s + E has a rightadjoint R . Moreover, for every object E ∈ E , evaluation at E commutes with this right adjoint,i.e., the square s + E s E s + Set sSet RR ev E ev E commutes (up to canonical isomorphism).Proof. We claim that for any X ∈ s + E , seen as a functor ∆ op+ → E , its right Kan extension along∆ op+ → ∆ op exists and is a pointwise right Kan extension. Indeed, the pointwise right Kan extensioncomputed at [ n ] ∈ ∆ should be RV = lim [ m ] → [ n ] ∈ E V ([ m ])where E is the comma category of [ m ] ∈ ∆ op+ endowed with a map [ m ] → [ n ] in ∆. This categoryis countable, so as E is countably complete, the limit exists, and hence the pointwise right Kanextension exists. By definition taking this right Kan extension is right adjoint to the forgetful functor s E → s + E , so this proves the existence of the right adjoint. The commutation of the square in theproposition is because the evaluation functor preserves limits, and hence preserves this pointwiseright Kan extension as well. Theorem 12.8. If E is countably complete, then both the forgetful functor and its right adjoint U : s E ⇆ s + E : R restrict to equivalences of fibration categories between s E fib and s + E fib .Proof. The theorem is valid for simplicial and semisimplicial sets, i.e., in the case of E = Set . Asboth U and R commute with evaluation at E ∈ E and weak equivalences and fibrations are detectedby these evaluations, it follows that: • U and R preserve fibrant objects and are morphisms of fibrations categories; • the unit and counit of the adjunctions are weak equivalences on fibrant objects.We now move to the case of a countably lextensive category E . Despite the fact that the theoremconcerns only the fibrant objects of s + E , the proof will depend on the homotopy theory of all, notnecessarily fibrant, semisimplicial objects in E . We define a general morphism of s + E to be a weakequivalence if it has a fibrant replacement (as constructed from factorisations of Lemma 12.1) that isa pointwise weak equivalence in s + E fib . This is analogous to the characterisation of weak equivalencesbetween simplicial objects in the model structure of Theorem 9.9. The weak equivalences, fibrations58nd cofibrations defined in this section do not form a model structure on s + E , but we can stillprove that they are sufficiently well-behaved for our purposes. For example, the definition of weakequivalences immediately implies that trivial cofibrations are weak equivalences. On the other hand,not all trivial fibrations are weak equivalences. Remark 12.9. If E is countably lextensive then s + E is a weak model category in the sense of [Hen18]with weak equivalences, fibrations and cofibrations as defined above. This can be derived from (thedual of) [Hen18, Proposition 2.3.3] and properties of the classes established in this section. In fact,as every object of s + E is cofibrant, this is even a right semi-model category, as long as we use thedefinition of a semi-model category in [Fre09] and not that in [Spi01] (see [Hen20, Section 3] forthe explanation of differences between the two definitions). Our discussion of homotopy theory ofsemisimplicial objects can be phrased both in terms of this weak model structure or right semi-model structure. However, we prefer to provide more elementary arguments to make this sectionmore self-contained. Proposition 12.10. If E has finite coproducts, then the forgetful functor s E → s + E has a leftadjoint. It is given by ( LX ) n = a [ n ] _ [ m ] X m where the coproduct is over all degeneracy operators [ n ] _ [ m ] in ∆ .Proof. The functor L is the left Kan extension along ∆ + → ∆. If it can be computed pointwise, itis given by the formula ( LX ) n = colim [ n ] → [ m ] X m where the colimit is taken over the comma category [ n ] ↓ ∆ op+ . (Its objects are arbitrary simplicialoperators [ n ] → [ m ], but its morphisms are just the face operators.) It follows from the existenceof the degeneracy/face unique factorisation system in ∆ that the discrete category of degeneracyoperators [ n ] _ [ m ] is cofinal in this category. Hence the colimit above can be rewritten as the thecoproduct in the statement of the proposition. Thus if E has finite coproducts, this colimit existswhich concludes the proof. Lemma 12.11.
The free functor L : s + E → s E preserves cofibrations and trivial cofibrations.Proof. It can be checked easily that the natural transformation from the initial functor to L satisfiesthe assumptions of Lemma 3.20, so it is enough to verify that L sends the generating cofibrationsand trivial cofibrations to cofibrations and trivial cofibrations, respectively. These generators areof the form Λ k + [ n ] ∆ + [ n ] or ∂ ∆ + [ n ] ∆ + [ n ] the image by L is computed as in Set , thus givingΛ k [ n ] ∆[ n ] or ∂ ∆[ n ] ∆[ n ], i.e., the generating cofibrations and trivial cofibrations in s E . Lemma 12.12.
The forgetful functor U : s E → s + E preserves cofibrations and trivial cofibrations.Proof. The forgetful functor preserves all colimits that exist so it is enough to show that thegenerating (trivial) cofibrations of s E are sent to (trivial) cofibrations. The case of cofibrationsfollows from Theorem 4.6 and Lemma 12.3. For trivial cofibrations, note that if X ∈ sSet , then U X = U X (the first U is the forgetful functor sSet → s + Set , the second one is s E → s + E ). Thusit is enough to show that U Λ n [ k ] → U ∆[ n ] is a trivial cofibration in s + E for all 0 ≤ k ≤ n . Forthis it is sufficient to show that Λ n [ k ] → U ∆[ n ] is a trivial cofibration in s + Set which was provenin [Hen18, Corollary 5.5.15 (ii)]. 59ote that the forgetful functor U preserves trivial fibrations, but trivial fibrations in s + E are notnecessarily weak equivalences. Nonetheless, the following statement is valid. Lemma 12.13.
The forgetful functor U : s E → s + E sends trivial fibrations to weak equivalences.Proof. This follows by the same argument as the second part of [Hen19, Lemma 2.2.1].
Lemma 12.14.
For each X ∈ s + E , the unit X → U LX is a trivial cofibration.Proof.
The composite
U L preserves all the relevant colimits, so it is enough to check that for eachgenerating cofibration ∂ ∆ + [ n ] → ∆ + [ n ], the map U L ( ∂ ∆ + [ n ]) ⊔ ∂ ∆ + [ n ] ∆ + [ n ] → U L ∆ + [ n ]is a trivial cofibration. It then follows from Lemma 3.20 that the same holds for all cofibrationsand the case of ∅ → X concludes the proof. Thus it suffices to prove the statement in the case ofsemisimplicial sets which is [Hen18, Proposition 5.5.14]. Proposition 12.15.
The forgetful functor U : s E → s + E preserves and reflects weak equivalences.Proof. The conclusion is valid for s E = sSet by [Hen19, Lemma 2.2.1] and thus it holds for mor-phisms between fibrant objects. Indeed, Hom s + Set ( E, U X ) = U Hom sSet ( E, X ) and weak equiva-lences between fibrant objects in both s E and s + E are detected by pointwise evaluation.For a general morphism X → Y , we consider its fibrant replacement as constructed in the smallobject argument. Since U preserves trivial cofibrations (by Lemma 12.12) and fibrations, it followsthat it preserves such fibrant replacements. Thus the conclusion follows from the special case ofmorphisms between fibrant objects. Corollary 12.16.
For each X ∈ s E , the counit LU X → X is a weak equivalence.Proof. This follows from the triangle identities using Lemma 12.14 and Proposition 12.15.
Theorem 12.17.
When E is countably lextensive, the functor U : s E fib → s + E fib is an equivalenceof fibration categories.Proof. Consider the functor L ′ : s + E fib → s E fib obtained by composing L with a chosen fibrantreplacement functor in s E . Such fibrant replacement along with the unit of the adjunction L ⊣ U induce a natural transformation id s + E fib → U L ′ which is a weak equivalence by Lemma 12.14and Proposition 12.15. Similarly, using the counit we obtain two natural transformations L ′ U X ← LU X → X for X ∈ s E . They are weak equivalences by definition and by Corollary 12.16, but LU is not an endofunctor of s E fib , just of s E . However, we can apply a functorial factorisation tothe morphism LU X → L ′ U X × X to obtain a weak equivalence LU X −→ ∼ T X and a fibration
T X ։ L ′ U X × X . Then T is an endofunctor of s E fib and we have two natural weak equivalences L ′ U ← T → id s E fib as required. Corollary 12.18.
Let E be a countably lextensive category. Then the effective model structure on s E is left proper.Proof. This follows by the combination of the following facts. First, the functor LU : s E → s E preserves colimits; secondly, LU preserves cofibrations by Lemmas 12.11 and 12.13; thirdly, LU takes values in cofibrant objects by Lemmas 12.3 and 12.11; and, finally, the counit LU X → X isa weak equivalence by Corollary 12.16. 60 ∞ -category Ho ∞ ( s E fib ) . Section 11 provides a description of the ∞ -category Ho ∞ s E presented by the effective modelstructure on s E when E is completely lextensive and locally connected. The goal of this section isto give an alternative characterisation of this ∞ -category under fewer assumptions on E . As shownin Section 1, if E is only a category with finite limits, we already have a fibration category structureon s E fib , which, in the case where E is countably lextensive corresponds to the category of fibrantobjects of the effective model structure hence models the same ∞ -category. We will consider themore general problem of describing the ∞ -category Ho ∞ s E fib in this case.We do not know such description for a general category E with finite limits, but we will presentan answer that applies when E is either countably complete or countably lextensive. More precisely,we will give a description of the ∞ -category Ho ∞ s + E fib , which we showed in Section 12 is equivalentto Ho ∞ s E fib when E is either countably lextensive or countably complete. Theorem 13.1.
Let E be a category that is either countably complete or countably lextensive.Then, evaluations at all E ∈ E induce a fully faithful embedding of Ho ∞ ( s E fib ) into the categoryof presheaves of spaces over E . More precisely, Ho ∞ ( s E fib ) is equivalent to the full subcategory ofpresheaves of spaces over E that are homotopy colimits (geometric realisations) of Kan complexesin E . This is closely related to the exact completion (or ex/lex completion) of E . In general, theexact completion (see, e.g., [CV98]) of a category E with finite limits can be described as the fullsubcategory of Psh E of objects that can be written as colimits of “setoids objects” in E , i.e., ascoequalisers of “proof-relevant equivalence relations”, that is diagrams R ⇒ X in E , such that theimage of the map Hom Set ( E, R ) → Hom
Set ( E, X ) × Hom
Set ( E, X )is an equivalence relation on Hom
Set ( E, X ) for each E ∈ E . The term “proof-relevant” refersto the fact that we do not assume that R → X × X is a monomorphism, or equivalently thatHom Set ( E, R ) is a subset of Hom
Set ( E, X ) × Hom
Set ( E, X ). The fact that R → X × X is a proof-relevant equivalence relation can be encoded as a structure consisting of morphisms in E witnessingtransitivity ( R × X R → R ), symmetry ( R → R ) and reflexivity ( X → R ). Proposition 1.4 canbe seen as a higher categorical version of this observation, i.e., Kan simplicial objects are a highercategorical generalisation of proof-relevant equivalence relations. In fact, it is easy to deduce fromthe theorem above that the full subcategory of set-truncated objects in Ho ∞ ( s E fib ) is equivalent tothe ex/lex completion of E .However, it does not seem accurate to think of Ho ∞ ( s E fib ) as the ∞ -categorical version of theex/lex completion. Let us say that an ∞ -category is exact if it has finite limits and quotients ofgroupoid objects exist and are van Kampen colimits. Lurie has shown that this condition togetherwith complete lextensivity and local presentability characterises ∞ -toposes [Lur09]. We can thendefine the ex/lex completion of an ∞ -category C with finite limits in the usual way: it is an exact ∞ -category C ex/lex with a functor C → C ex / lex such that any finite limit preserving functor toan exact ∞ -category C → D extends essentially uniquely to an exact functor C ex/lex → D . Weconjecture that the effective model structure is related to this ex/lex completion operation in thefollowing way: Conjecture 13.2.
Let E be a countably lextensive category or countably complete category. Theex/lex completion of the ∞ -category associated to E is equivalent to the full subcategory of Ho ∞ ( s E fib ) on objects that are n -truncated for some n . E when Ho ∞ ( s E fib )is replaced with Ho ∞ ( s + E fib ).The general idea of the proof of Theorem 13.1 is that the category Fam E of families of objectsof E is always a completely lextensive locally connected category, such that E can be identified withits category of connected objects. Hence, we can apply Theorem 11.7 to it and show thatHo ∞ ( sFam E ) ≃ Ho ∞ sPsh E The right hand side is a model for the ∞ -category of small presheaves of spaces on E (in the ∞ -categorical sense). We always have a fully faithful embedding s E → sFam E which identifies s E withthe full subcategory of levelwise connected simplicial objects. Moreover, a map is a fibration or aweak equivalence (between fibrant objects) in s E if and only if its image in sFam E is one, so thisembedding also restricts to a morphism of fibration categories.Our goal is to show that (under the assumptions of Theorem 13.1) this also induces a fully faithfulembedding on the level of the ∞ -categories. Unfortunately, we are able to give a proof of this onlywhen we consider instead the semisimplicial version of this embedding s + E → s + Fam E . But as Fam E is always countably lextensive we have an equivalence of fibration categories sFam E ≃ s + Fam E by Theorem 12.17, and as soon as E is countably complete or countably lextensive we have anequivalence Ho ∞ ( s + E fib ) ≃ Ho ∞ ( s E fib ) by Theorem 12.6. So we need to show that s + E → s + Fam E induces a fully faithful functor between the corresponding ∞ -categories. Because of the followinglemma, it is enough to prove that it is fully faithful at the level of the homotopy categories. Lemma 13.3.
A finite limit preserving functor between two ∞ -categories which is an equivalence(resp. fully faithful) on the homotopy categories is an equivalence (resp. fully faithful).Proof. This is shown for the case of equivalences in [Cis20, Theorem 7.6.10]. The case of fullyfaithful functors can be deduced from the case of equivalences. Let f : X → Y be a finite limitpreserving functor which is fully faithful on the homotopy category and let Y ′ denote its essentialimage. Then Y ′ contains the terminal object since f preserves finite limits. Similarly, Y ′ is closedunder pullbacks. Indeed, since f is fully faithful on the homotopy categories, any cospan in Y ′ can be lifted to a cospan in X . Its pullback exists in X and is preserved by f . It follows that f induces a finite limit preserving functor X → Y ′ which is fully faithful and essentially surjectiveon the homotopy categories, so it is an equivalence, and hence by the result above, f induces anequivalence between X and Y ′ , i.e., it is fully faithful. Theorem 13.4.
For any category E with finite limits, the functor s + E fib → ( s + Fam E ) fib is fullyfaithful on the homotopy categories.Proof. The homotopy category of s + E fib is the quotient by the homotopy relation defined via maps X → ∆ + [1] ⋔ Y . This follows since all semisimplicial objects are cofibrant and s + E fib is a pathcategory in the sense of [BM18a]. The functor s + E fib → ( s + Fam E ) fib preserves finite limits and henceit preserves cotensors by ∆ + [1]. Thus morphisms in s + E fib are homotopic in s + E fib if and only if theyare homotopic in s + Fam E fib . Remark 13.5.
The crucial difference between semisimplicial and simplicial settings is that everysemisimplicial object in s + E is cofibrant in s + Fam E . However, a non-constant simplicial object in s E is levelwise connected in sFam E and thus not cofibrant by Theorem 4.6.We are now ready to prove Theorem 13.1. 62 roof of Theorem 13.1. We always have a diagram of functors: s E fib s + E fib ( sFam E ) fib ( s + Fam E ) fib Theorem 12.17 shows that the bottom horizontal functor is always an equivalence of the ho-motopy categories as
Fam C is always a completely lextensive category. The top horizontal map isalso an equivalence on the homotopy categories by Theorem 12.6 since E is countably lextensiveor countably complete. Finally, we have shown in Theorem 13.4 that the right vertical functoris fully faithful on the homotopy categories. It follows that the left vertical functor is also fullyfaithful on the homotopy categories, and hence by Lemma 13.3 induces a fully faithful embeddingof ∞ -categories s E fib → ( sFam E ) fib .Now, Fam E is a locally connected completely lextensive category, and E is its category ofconnected objects. Hence, by Theorem 11.7, the ∞ -category Ho ∞ ( sFam E ) fib is equivalent to thecategory of presheaves of spaces over E , which proves the first half of the theorem.For the description of the essential image we simply investigate the precise nature of the em-bedding constructed above. If X ∈ s E fib then its image in ( sFam E ) fib is also fibrant, and the objectscorresponding to E ∈ E are cofibrant, so, as this is a simplicial model category, the Hom spacein the corresponding ∞ -category between them is simply Hom sSet ( E, X ). Hence X is sent to thepresheaf of spaces E Hom sSet ( E, X ). Note that as colimits in presheaf categories are computedlevelwise and the colimit of a simplicial set in the ∞ -category of spaces is the the spaces representedby this simplicial sets, this can equivalently be expressed as the fact that X is sent to its geometricrealisation in the presheaf category. Appendix A Remarks on constructivity
While the present paper has been written within ZFC for simplicity, many of our results andproofs are constructive, i.e., do not rely on the law of excluded middle or the axiom of choice,subject to some clarifications, which we will discuss briefly here.First of all, in the constructive reading of the paper, a finite set means a finite cardinal, or afinite decidable set, i.e., a set equipped with a bijection with { , . . . , n } , for some n ∈ N . A countableset is a set which is equipped with a bijection with either { , . . . , n } or N . With this definition, acountable coproduct of countable sets is countable.Secondly, we restrict ourselves to consider finitely lextensive, countably lextensive and com-pletely lextensive categories. Here, by a finitely lextensive category we mean a category with astrict initial object and van Kampen binary coproducts and by a countably lextensive category wemean a finitely lextensive category that in addition has N -indexed van Kampen coproducts. Withthis definition, the category of countable sets is countably lextensive. Without these changes, wewould run into problems as ω is not a regular cardinal in ZF and the axiom of countable choice isneeded to show that a countable union of countable set is countable, and therefore Definition 2.7would be problematic, as we could not show that the category of countable sets is countably lex-tensive.Finally, one should assume the convention that every time we discuss existence of an object,this involves explicit structure, rather than a mere property. For example, when we say that a63ap f has the left lifting property against g , we mean that f comes equipped with a function thatassociates a solution to each lifting problem.We make no claim on whether it is possible to make the results in Section 11 and Section 13constructive. Indeed, both of these sections involve ∞ -categories, for which a constructive theoryhas not been developed yet. Also, Lemma 11.3 is non-constructive: in a constructive setting, itsconclusion should be taken as the definition of a connected object. Finally, Section 11 relies on theexistence of the projective model structure on the category of small presheaves on a large category,which is not known to exist constructively. References [AW09] S. Awodey and M. Warren,
Homotopy-theoretic models of identity types , Math. Proc. Camb. Phil. Soc. (2009), no. 1, 45–55.[Bar10] C. Barwick,
On left and right model categories and left and right Bousfield localizations , Homology, Ho-motopy and Applications (2010), no. 2, 245–320.[BM18a] B. van den Berg and I. Moerdijk, Exact completion of path categories and algebraic set theory: Part I:Exact completion of path categories , Journal of Pure and Applied Algebra (2018), no. 10, 3137–3181.[BM18b] B. van den Berg and I. Moerdijk,
Univalent completion , Mathematische Annalen (2018), 1337–1350.[Bro73] K. S. Brown,
Abstract homotopy theory and generalized sheaf cohomology , Trans. Amer. Math. Soc. (1973), 419–458.[BR13] J. Bergner and C. Rezk,
Reedy categories and the Θ -construction , Mathematische Zeitschrift (2013),no. 1-2, 499–514.[CLW93] A. Carboni, S. Lack, and R. F. C. Walters, Introduction to extensive and distributive categories , J. PureAppl. Algebra (1993), no. 2, 145–158.[CV98] A. Carboni and E. M. Vitale, Regular and exact completions , J. Pure Appl. Algebra (1998), no. 1–3,79–116.[Car95] A. Carboni,
Some free constructions in proof theory and realizability , J. Pure Appl. Algebra (1995),no. 2, 117-148.[CD09] B. Chorny and W. G. Dwyer,
Homotopy theory of small diagrams over large categories , Forum Mathe-maticum, 2009, pp. 167–179.[Cho15] B. Chorny,
Homotopy theory of relative simplicial presheaves , Israel Journal of Mathematics (2015),no. 1, 471–484.[Cis20] D.-C. Cisinski,
Higher categories and homotopical algebra , Cambridge Studies in Advanced Mathematics,Cambridge University Press, 2020.[CH02] J. D. Christensen and M. Hovey,
Quillen model structures for relative homological algebra , Math. Proc.Cambridge Philos. Soc. (2002), no. 2, 261-293.[DL07] B. J. Day and S. Lack,
Limits of small functors , Journal of Pure and Applied Algebra (2007), no. 3,651–663.[DHI04] D. Dugger, S. Hollander, and D. C. Isaksen,
Hypercovers and simplicial presheaves , Math. Proc. Camb.Phil. Soc. (2004), no. 1, 9–51.[DK84] W. G. Dwyer and D. M. Kan,
Singular functors and realization functors , Indagationes Mathematicae(Proceedings), 1984, pp. 147–153.[Elm83] A. D. Elmendorf,
Systems of fixed point sets , Transactions of the American Mathematical Society (1983), no. 1, 275–284.[EP17] J. Emmenegger and E. Palmgren,
Exact completion and constructive theories of sets (2017), available at https://arxiv.org/abs/1710.10685 . To appear in Journal of Symbolic Logic.[F87] E. Dror Farjoun,
Homotopy theories for diagrams of spaces , Proceedings of the American MathematicalSociety (1987), no. 1, 181–189. Fre09] B. Fresse,
Modules over operads and functors , Lecture Notes in Mathematics. Springer-Verlag, Berlin (2009).[GZ67] P. Gabriel and M. Zisman,
Calculus of fractions and homotopy theory , Ergebnisse der Mathematik undihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967.[GS17] N. Gambino and C. Sattler,
The Frobenius property, right properness and uniform fibrations , J. PureAppl. Algebra (2017), No. 12, 3027–3068.[GK17] D. Gepner and J. Kock,
Univalence in locally Cartesian closed ∞ -categories , Forum Math. (2017),617–652.[GSS19] N. Gambino, C. Sattler, and K. Szumi lo, The constructive Kan–Quillen model structure: two new proofs (2019), available at https://arxiv.org/abs/1907.05394 .[Gar09] R. Garner,
Understanding the small object argument , Appl. Cat. Struct. (2009), no. 3, 247–285.[GJ99] P. Goerss and R. Jardine, Simplicial homotopy theory , Birkauser, 1999.[Hen18] S. Henry,
Weak model categories in classical and constructive mathematics (2018), available at https://arxiv.org/abs/1807.02650 .[Hen19] S. Henry,
A constructive account of the Kan–Quillen model structure and of Kan’s Ex ∞ functor (2019),available at https://arxiv.org/abs/1905.06160 .[Hen20] S. Henry, Combinatorial and accessible weak model categories (2020), available at https://arxiv.org/abs/2005.02360 .[Hof97] M. Hofmann,
Extensional concepts in intensional type theory , Springer, 1997.[Hyl82] M. Hyland,
The effective topos , The L. E. J. Brouwer Centenary Symposium, 1982, pp. 165-216.[HT96] H. Hu and W. Tholen,
A note on free regular and exact completions and their infinitary generalizations ,Theor. App. Cat. (1996), no. 10, 113-132.[Joh02] P. T. Johnstone, Sketches of an elephant: a topos theory compendium. Vol. 1 , Oxford Logic Guides, vol. 43,The Clarendon Press, Oxford University Press, New York, 2002. MR1953060[Jar96] R. Jardine,
Boolean localisation in practice , Documenta Mathematica (1996), 245–275.[Joy84] A. Joyal, Letter to A. Grothendieck (1984), available at https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf .[JT07] A. Joyal and M. Tierney,
Quasi categories vs Segal spaces , Categories in algebra, geometry and mathe-matical physics, 2007, pp. 277-326.[KL12] C. Kapulkin and P. LeFanu Lumsdaine,
The Simplicial Model of Univalent Foundations (after Voevodsky) (2012), available at https://arxiv.org/abs/1211.2851 .[Lur09] J. Lurie,
Higher topos theory , Princeton University Press, 2009.[MLM92] S. Mac Lane and I. Moerdijk,
Sheaves in geometry and logic - A first introduction to topos theory , Springer,1992.[Men03] M. Menni,
A characterization of the left exact categories whose exact completions are toposes , Journal ofPure and Applied Algebra (2003), no. 3, 287–301.[MV99] F. Morel and V. Voevodsky, A -homotopy theory of schemes , Publ. Math. I.H.E.S (1999), 45-143.[Qui67] D. G. Quillen, Homotopical algebra , Lecture Notes in Mathematics, vol. 43, Springer, 1967.[RB06] A. R˘adulescu-Banu,
Cofibrations in Homotopy Theory (2006), available at https://arxiv.org/abs/math/0610009v4 .[Ras18] N. Rasekh,
A Theory of Elementary Higher Toposes (2018), available at https://arxiv.org/abs/1805.03805 .[Rez01] C. Rezk,
A model for the homotopy theory of homotopy theories , Transactions of the American Mathe-matical Society (2001), no. 3, 973–1007.[Rez10] C. Rezk,
Toposes and Homotopy Toposes (2010), available at https://faculty.math.illinois.edu/~rezk/homotopy-topos-sketch.pdf .[Rie14] E. Riehl,
Categorical Homotopy Theory , Cambridge University Press, 2014.[RV14] E. Riehl and D. Verity,
The theory and practice of Reedy categories , Theory Appl. Categ. (2014),256–301. Sat17] C. Sattler,
The Equivalence Extension Property and Model Structures (2017), available at https://arxiv.org/abs/1704.06911 .[Shu17] M. Shulman,
Elementary ( ∞ , -topoi (2017), available at https://golem.ph.utexas.edu/category/2017/04/elementary_1topoi.html .[Shu19] M. Shulman, All ( ∞ , -toposes have strict univalent universes (2019), available at https://arxiv.org/abs/1904.07004 .[Spi01] M. Spitzweck, Operads, Algebras and Modules in General Model Categories (2001), available at https://arxiv.org/abs/math/0101102 .[Ste17] W. Steimle,
Degeneracies in quasi-categories (2017), available at https://arxiv.org/abs/1702.08696 .[Ste16] M. Stephan,
On equivariant homotopy theory for model categories , Homology, Homotopy and Applications (2016), no. 2, 183–208.[Szu17] K. Szumi lo, Homotopy theory of cocomplete quasicategories , Algebraic & Geometric Topology (2017),765–791.[TV05] B. To¨en and G. Vezzosi, Homotopical algebraic geometry I: higher topos theory , Advances in Mathematics (2005), 257–372.
N. Gambino,
University of Leeds , [email protected]
S. Henry,
University of Ottawa , [email protected] C. Sattler,
Chalmers University of Technology , [email protected] K. Szumi lo,