aa r X i v : . [ m a t h . C T ] F e b CARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES
NIMA RASEKHA
BSTRACT . Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are com-monly used in ( ∞ , 1 ) -category theory to study presheaves valued in ( ∞ , 1 ) -categories. In this work we define andstudy fibrations modeling presheaves valued in simplicial spaces and their localizations. This includes defininga model structure for these fibrations and giving effective tools to recognize its fibrations and weak equivalences.This in particular gives us a new method to construct Cartesian fibrations via complete Segal spaces. In additionto that, it allows us to define and study fibrations modeling presheaves of Segal spaces. C ONTENTS
0. Introduction 11. Reviewing Concepts 52. The Reedy Covariant Model Structure 103. Localizations of Reedy Left Fibrations 134. (Segal) Cartesian Fibrations 23References 25I
NTRODUCTION
The theory of ( ∞ , 1 ) -categories helped formalize the notionof homotopies that first arose in classical algebraic topology. This helped overcome many early challenges inalgebraic topology. For example it helped develop a homotopy invariant notion of colimit, making senseof homotopy colimits [BK72], or helped properly define a smash product of spectra [GGN15], an importantproblem in the early days of stable homotopy theory [Ada95, EKMM95]. More generally, it created a foun-dation for properly developing “homotopy coherent mathematics” , which has now found applications in manybranches of mathematics, such as algebraic geometry [Lur18], differential geometry [NSS15a, NSS15b] andeven mathematical physics [Lur09c]. As one might expect such benefits also come with a price. For exam-ple, it greatly complicates the notion of functoriality, which now needs to be homotopy coherent and hencerequires checking an infinite number of conditions.Fortunately, certain important classes of functors can be defined in alternative ways, that are often easierto construct in practice. For example functors out of an ( ∞ , 1 ) -category into the ( ∞ , 1 ) -category of spacesare equivalent to left fibrations over that ( ∞ , 1 ) -category. This was first observed by Joyal who was develop-ing the category theory of quasi-categories , a popular model of ( ∞ , 1 ) -categories [Joy08a, Joy08b]. It was thenfurther studied by Lurie, who also presented one of the first proofs of the equivalence between functors andfibrations in the context of quasi-categories [Lur09a]. In the subsequent years many authors have reviewedthe theory of left fibrations and its relation with functors from many different perspectives: There are al-ternative methods for defining the model structure for left fibrations, the covariant model structure , in the Date : February 2021. context of quasi-categories [Ngu19]. Moreover, there are now many alternative proofs of the equivalencebetween left fibrations and functors again in the quasi-categorical context [HM15, HM16, Ste17, Cis19].There are also studies of left fibrations using complete Segal spaces [Rez01], another model of ( ∞ , 1 ) -categories[BdB18, Ras17b, KV14]. Moreover, there is an analysis of left fibrations in the context of an ∞ -cosmos, whichis a model-independent approach to ( ∞ , 1 ) -category theory using various ideas from 2-category theory[RV21]. Finally, left fibrations have also been studied in a homotopy type theoretical context [RS17].Another class of functors that can studied via fibrations are functors valued in ( ∞ , 1 ) -categories them-selves. Here the corresponding fibrations are known as coCartesian fibrations . These were first definedby Lurie [Lur09a], who proved an equivalence between fibrations and functors by constructing a Quillenequivalence between appropriately defined model categories. However, coCartesian fibrations have notreceived the same attention that left fibrations have. There has been interesting work on the model-independent aspects of coCartesian fibrations, both from a quasi-categorical perspective [MG19, AF20] aswell as from an ∞ -cosmos perspective [RV21]. However, the coCartesian model structure and its equivalencewith functors in [Lur09a] have not been tackled again in the quasi-categorical setting, let alone other modelsof ( ∞ , 1 ) -categories.There are several complicating factors that have contributed to our current predicament. One very mys-terious issue that arises when studying coCartesian fibrations is that although quasi-categories are simpli-cial sets, the model structure for coCartesian fibrations has only been defined for marked simplicial sets and itis widely believed that it is not possible to define an appropriate model structure on simplicial sets that canhelp us study coCartesian fibrations. This technicality adds a layer, in particular as the category of markedsimplicial sets is not a presheaf category hence depriving us of many techniques to study fibrations (that forexample play an important role in [Cis19, Ngu19]). Another complicating factor comes from the fact thatfunctors into ( ∞ , 1 ) -categories have an inherent ( ∞ , 2 ) -categorical character (as we can talk about naturaltransformations of such functors) and while there are several models of ( ∞ , 2 ) -categories in the literature[Rez10, Bar05, Ver08, Ara14], the study of its category theory and in particular fibrations is still in its earlystages [Lur09b, GHL20]. Up to this point we discussed the importance ofcoCartesian fibrations and the need to find alternative perspectives. The goal of this paper is to offer onesuch alternative perspective using complete Segal objects (also called
Rezk objects [RV17]). Before going intofurther details it is instructive to review the construction of complete Segal spaces, due to Rezk [Rez01],which goes as follows:(1) He starts with the category of simplicial sets with the Kan model structure, giving us a model forspaces.(2) He then takes simplicial diagrams X in spaces, defining simplicial spaces , and gives that the Reedymodel structure.(3) He adds two restrictions by using Bousfield localization on the Reedy model structure:(I) Segal Condition:
A Reedy fibrant simplicial space X is a Segal space if the map X n → X × X ... × X X for n ≥ Completeness Condition:
A Segal space X is a complete Segal space if the map X hoequiv → X is a Kan equivalence, where X hoequiv can be described as a finite limit [Rez10, 10].For a review of complete Segal spaces see Subsection 1.5.Later complete Segal spaces model structure was proven to be equivalent to the model structure for quasi-categories [JT07] and other models of ( ∞ , 1 ) -categories [To¨e05, Ber10].The beauty of the complete Segal space approach to ( ∞ , 1 ) -categories is that the process we outlinedabove can be generalized from spaces to any ( ∞ , 1 ) -category with finite limits, giving us a notion of com-plete Segal objects. In particular we can apply the process to the ( ∞ , 1 ) -category of functors valued in ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 3 spaces, or, equivalently, to left fibrations. Applying the process to the ( ∞ , 1 ) -category of space-valued func-tors evidently results in the ( ∞ , 1 ) -category of ( ∞ , 1 ) -category-valued functors. This naturally motivatesstudying complete Segal objects of left fibrations as a fibrational analogue to functors valued in ( ∞ , 1 ) -categories hence suggest following approach to coCartesian fibrations: coCartesian fibrations are complete Segal objects in left fibrations.0.3 Cartesian Fibration of Complete Segal Spaces. In order to start the process from left fibrations tocoCartesian fibrations we first need to choose a model for our left fibrations. Here we will use left fibrationsof simplicial spaces as studied in [Ras17b], however, it should be noted that using left fibrations as studiedby Lurie [Lur09a] would give us the same results. In fact the equivalence of the two resulting coCartesianmodel structures has been proven in the follow-up work [Ras21].Having decided which model of left fibrations to use we define coCartesian fibrations and study theirproperties simply by following the same three steps that Rezk used:(1) Start with the category of simplicial spaces over a fixed simplicial space and give it a model structuresuch that the fibrant objects are the left fibrations. This model structure is known as the covariantmodel structure [Ras17b] and is reviewed in Subsection 1.6.(2) Take simplicial diagrams in left fibrations. Then give the resulting category a Reedy model struc-ture, calling it the
Reedy covariant model structure on bisimplicial spaces . Then observe how the prop-erties of the covariant model structure transfers to the Reedy covariant model structure. This is thecontent of Section 2.(2.5) Next we will do a general analysis how the Reedy covariant model structure behaves when we useBousfield localizations, in particular analyzing the fibrant objects (Corollary 3.16) and weak equiv-alences (Theorem 3.20) and studying its invariance (Theorem 3.21). This is the goal of Section 3.(3) Finally, we apply the results of the previous section and focus on the particular conditions associatedwith Segal spaces and complete Segal spaces to define Segal coCartesian fibrations and coCartesianfibrations. We will cover that in Section 4.Note we can prove that the resulting model structure is Quillen equivalent to the Cartesian model structureon marked simplicial sets defined by Lurie. That is the main result of the follow up work [Ras21].
Given that we already had a Cartesian model structure, whypresent an alternative way? Beside a theoretical satisfaction of approaching an interesting topic from a newangle, there are also concrete benefits:(1)
Exposition:
Cartesian fibrations are notoriously difficult to understand. The main source for manyresults is still [Lur09a, Chapter 3] and is quite technical, requiring a lot of background knowledge.This makes it difficult for most, except for a small number of experts, to use Cartesian fibrations toprove new results.The complete Segal object approach to Cartesian fibrations requires far less theoretical back-ground. It primarily relies on understanding left fibrations, which are in fact easier and have beenstudied by many different people meaning there are now excellent resources for mathematiciansinterested in fibrations.(2)
Direct Proofs:
One important implication of the equivalence between left fibrations and space-valued functors is the fact that the covariant model structure is invariant under ( ∞ , 1 ) -categoricalequivalences, meaning that an equivalence in the model structure for quasi-categories gives us aQuillen equivalence of covariant model structures [Lur09a, Remark 2.1.4.11]. However, there arenow also direct proofs of this fact that only use structural properties of the covariant model struc-ture, both in the setting of quasi-categories [HM15] as well as complete Segal spaces [Ras17b].Generalizing to coCartesian fibrations, we can still use the equivalence with functors to deduce itis invariant under ( ∞ , 1 ) -categorical equivalences, as has been done in [Lur09a, Proposition 3.3.1.1].However, we do not have a direct proof only using the coCartesian model structure on markedsimplicial sets. On the other hand, using the complete Segal approach to coCartesian fibrations NIMA RASEKH allows us to generalize the invariant proof for left fibrations to coCartesian fibrations in a reasonablystraightforward manner (as shown in Theorem 4.12).(3)
Segal coCartesian Fibrations:
Homotopy type theory is a new approach to the foundations that is in-herently homotopy invariant [Uni13]. It has opened the possibility of finding a model independentapproach to ( ∞ , 1 ) -category theory. As one would expect of an axiomatic system, one importantquestion is their expressiveness, meaning which axioms are necessary to prove which result. Animportant example is the univalence axiom which is simply the type theoretic articulation of thecompleteness condition we use to define complete Segal spaces. For example, in their paper [RS17]Riehl and Shulman introduce a notion of ( ∞ , 1 ) -category, a Rezk type , inside their type theory. Theythen prove that the Yoneda lemma holds without the univalence axiom, whereas equality of variousnotions of adjunctions does require univalence.In the language of ( ∞ , 1 ) -categories trying to tackle this question corresponds to proving a re-sult for a general Segal spaces vs. observing that the completeness condition is in fact necessary.For example, the independence of univalence from the Yoneda lemma corresponds to proving theYoneda lemma for Segal spaces, which in fact has been done (independently) in [Ras17b]. In orderto seriously pursue such question we would need a notion of fibration for Segal spaces, similar tocoCartesian fibrations. Defining such fibrations does not seem possible using marked simplicialsets, whereas we can do so using the complete Segal approach (Section 4). Hence, using the com-plete Segal approach to coCartesian fibration allows us to tackle more general question of interestrelated to foundations and the necessity of completeness.(4) Representable Cartesian Fibrations:
One important class of left fibrations are representable left fi-brations , which are precisely the fibrations that correspond to corepresentable functors. These leftfibrations play an extraordinary role in ( ∞ , 1 ) -category theory and many important results (such aslimits, adjunctions, ...) can be reduced to determining the representability of certain left fibrations.We can similarly try to determine when a coCartesian fibration is representable by a simplicialobject. While it is theoretically possible to study such coCartesian fibrations using the marked sim-plicial approach (as has been done in [Ste20]), the complete Segal approach is perfectly tailoredto tackle such questions. The study of such representable coCartesian fibrations deserves its ownattention and hence is part of a follow up to this paper [Ras17a].(5) Fibrations of ( ∞ , n ) -Categories: The same way that ( ∞ , 1 ) -category theory has led to a precise no-tion of “weak 1-categories”, the development of ( ∞ , n ) -categories is helping us conceptualize weakn-categories . Though in its early stages it has already contributed to the advancement of topologi-cal field theories [Lur09c, CS19], derived algebraic geometry [GR17a, GR17b] and ( ∞ , 1 ) -categorytheory itself [Lur09b]. Further applications and studies require a good theory of fibrations.Some common models of ( ∞ , n ) -categories, such as Θ n -spaces [Rez10] and n -fold complete Segalspaces [Bar05] are in fact direct (and equivalent [BR13, BR20]) generalizations of complete Segalspaces. As the complete Segal object to fibrations is inherently inductive, it opens the possibilityof defining fibrations for ( ∞ , n ) -categories by simply choosing appropriate Θ n -diagrams or n -foldsimplicial diagrams in left fibrations. This paper is the first part of a three-paper series which introduces the bisim-plicial approach to Cartesian fibrations:(1)
Cartesian Fibrations of Complete Segal Spaces (2)
Quasi-Categories vs. Segal Spaces: Cartesian Edition [Ras21](3)
Cartesian Fibrations and Representability [Ras17a]In particular, the second paper proves that the approach here coincides with the approach via marked sim-plicial sets. The third paper gives an application of the bisimplicial approach to the study of representableCartesian fibrations.
I want to thank my advisor Charles Rezk who suggested this topic to me. I alsowant to thank the referee of the paper [Ras17b] for suggestions that have also resulted in many improve-ments of this work.
ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 5 R EVIEWING C ONCEPTS
In this section we review some basic concepts regarding model categories, simplicial spaces, completeSegal spaces, left fibrations and bisimplicial spaces that we will need in the coming sections.
We will use the language of model categories throughout and so use results from[Hov99, Hir03, Lur09a, JT07]. Here we will only state few results explicitly.
Remark . Recall a model structure M on a category C is called compatible with Cartesian closure if forcofibrations i , j and fibration p , the pushout-product i (cid:3) j is a cofibration and the pullback-exponential exp ( i , p ) is a fibration, which is trivial if either maps involved are trivial.For more details pushout products and pullback exponentials and their interaction (also known as Joyal-Tierney calculus ) see the original source [JT07, Section 7] or [Ras17b, Subsection 2.1]. We also need a resultguaranteeing that Bousfield localizations preserve Quillen equivalences.
Theorem 1.2.
Let C and be D two categories and M and N two simplicial, combinatorial model structures suchthat the cofibrations are inclusions in C and D respectively. Moreover, C M D N F ⊥ G be a simplicial Quillen adjunction (equivalence) of model structures and S a set of cofibrations in M . Then we get aQuillen adjunction (equivalence) C M S D N F ( S ) F ⊥ G where the left hand side is the localized model structure with respect to S and the right hand side has been localizedwith respect to F ( S ) .Proof. First we assume ( F , G ) is a Quillen adjunction between the M and N model structure and prove it isa Quillen adjunction between the M S and N F ( S ) model structure. We know that F preserves cofibrations,hence, by [Lur09a, Corollary A.3.7.2] it suffices to check that the right adjoint G preserves fibrant objects.Let Y in D be N F ( S ) -fibrant. Then GY is M -fibrant and so we only need to prove that for all maps f : A → B in S (1.3) f ∗ : Map C ( B , GY ) → Map C ( A , GY ) is a Kan equivalence. By adjunction this is equivalent to(1.4) F ( f ) ∗ : Map D ( FB , Y ) → Map D ( FA , Y ) being an equivalent, which holds by assumption. Notice, we can use the same argument to deduce that ifan object Y in D is N -fibrant, such that G ( Y ) is M S -fibrant, then Y is in fact N F ( S ) -fibrant. Indeed, N F ( S ) -fibrancy implies the map 1.3 is an equivalence which implies that 1.4 is an equivalence giving us the desiredresult.Next we want to prove that if ( F , G ) is a Quillen equivalence between the M and N -model structuresthen it is also a Quillen equivalence between the M S and N F ( S ) model structure. First, observe the derivedcounit map is an equivalence. Indeed, all objects are cofibrant, which means the derived counit map is justthe counit map, which by assumption is an equivalence in N and hence in N F ( S ) .Next we show the derived unit map is an equivalence. Let X be an M F ( S ) -bifibrant object in C . Let R ( F ( X )) be an N -fibrant replacement of F ( X ) . Then R ( F ( X )) is in fact N F ( S ) -fibrant and hence an N F ( S ) -fibrant replacement. Indeed, by the previous paragraph it suffices to prove that G ( R ( F ( X )) is M S -fibrant.However, as ( F , G ) is a Quillen equivalence between the M and N model structure, it is equivalence to X NIMA RASEKH (via the derived unit map) and hence is M S -fibrant by assumption. Hence X → G ( R ( F ( X ))) is in fact thederived unit map in the M S model structure. By assumption it is an M -equivalence and so it is also an M S -equivalence, finishing the proof. (cid:3) S will denote the category of simplicial sets, which we will call spaces . We will usethe following notation with regard to spaces:(1) ∆ is the indexing category with objects posets [ n ] = {
0, 1, ..., n } and mappings maps of posets.(2) We will denote a morphism [ n ] → [ m ] by a sequence of numbers < a , ..., a n > , where a i is the imageof i ∈ [ n ] .(3) ∆ [ n ] denotes the simplicial set representing [ n ] i.e. ∆ [ n ] k = Hom ∆ ([ k ] , [ n ]) .(4) ∂ ∆ [ n ] denotes the boundary of ∆ [ n ] i.e. the largest sub-simplicial set which does not include id [ n ] : [ n ] → [ n ] .(5) Let I [ l ] be the category with l objects and one unique isomorphisms between any two objects. Thenwe denote the nerve of I [ l ] as J [ l ] . It is a Kan fibrant replacement of ∆ [ l ] and comes with an inclusion ∆ [ l ] J [ l ] , which is a Kan equivalence. s S = Fun ( ∆ op , S ) denotes the category of simplicial spaces (bisimplicial sets). Wehave the following basic notations with regard to simplicial spaces:(1) We embed the category of spaces inside the category of simplicial spaces as constant simplicialspaces (i.e. the simplicial spaces S such that S n = S for all n and all simplicial operator maps areidentities).(2) More generally we say a simplicial space is homotopically constant if all simplicial operator maps X n → X m are equivalences (and in particular X n are all equivalent to X ).(3) Denote F ( n ) to be the simplicial space defined as F ( n ) kl = ∆ [ n ] k = Hom ∆ ([ k ] , [ n ]) . Moreover ∂ F [ n ] denotes the boundary of F ( n ) .(4) Denote E ( n ) to be the simplicial space defined as E ( n ) kl = J [ n ] k , as defined in Subsection 1.2(5).(5) The category s S is enriched over spacesMap s S ( X , Y ) n = Hom s S ( X × ∆ [ n ] , Y ) .(6) The category s S is also enriched over itself ( Y X ) kn = Hom s S ( X × F ( n ) × ∆ [ l ] , Y ) .(7) By the Yoneda lemma, for a simplicial space X we have a bijection of spaces X n ∼ = Map s S ( F ( n ) , X ) . The category of simplicial spaces has a Reedy model structure [Ree74],which is defined as follows:(F) A map f : Y → X is a (trivial) Reedy fibration if for each n ≥ s S ( F ( n ) , Y ) → Map s S ( ∂ F ( n ) , Y ) × Map s S ( ∂ F ( n ) , X ) Map s S ( F ( n ) , X ) .(W) A map f : Y → X is a Reedy equivalence if it is a level-wise Kan equivalence.(C) A map f : Y → X is a Reedy cofibration if it is a monomorphism.The Reedy model structure is very helpful as it enjoys many features that can help us while doing compu-tations. In particular, it is combinatorial , simplicial and proper . Moreover, it is also compatible with Cartesianclosure (Remark 1.1). These properties in particular imply that we can apply Bousfield localizations to theReedy model structure. See [Hir03] for more details. ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 7
The Reedy model structure can be localized such that it models ( ∞ , 1 ) -categories [Rez01]. This is done in two steps. First we define Segal spaces . A Reedy fibrant simplicialspace X is called a Segal space if the mapMap ( F ( n ) , X ) ≃ −−→ Map ( G ( n ) , X ) induced by the spine inclusion G ( n ) ֒ → F ( n ) is a Kan equivalence for n ≥ Theorem 1.5. [Rez01, Theorem 7.1]
There is a unique combinatorial left proper simplicial model structure on thecategory s S of simplicial spaces called the Segal space model category structure, and denoted s S Seg , with the followingproperties.(1) The cofibrations are the monomorphisms.(2) The fibrant objects are the Segal spaces.(3) The weak equivalences are the maps f such that
Map s S ( f , W ) is a weak equivalence of spaces for every Segalspace W. Segal spaces do not give us a model of ( ∞ , 1 ) -categories. For that we need complete Segal spaces . A Segalspace is called a complete Segal space if the mapMap ( E ( ) , W ) → Map ( F ( ) , W ) induced by the inclusion F ( ) → E ( ) (Subsection 1.3(4)) is a Kan equivalence. Complete Segal spacescome with their own model structure, the complete Segal space model structure . Theorem 1.6. [Rez01, Theorem 7.2]
There is a unique combinatorial left proper simplicial model structure on thecategory s S of simplicial spaces, called the complete Segal space model category structure, and denoted s S CSS , withthe following properties.(1) The cofibrations are the monomorphisms.(2) The fibrant objects are the complete Segal spaces.(3) The weak equivalences are the maps f such that
Map s S ( f , W ) is a weak equivalence of spaces for everycomplete Segal space W. A complete Segal space is a model for an ( ∞ , 1 ) -category. For a better understanding of complete Segalspaces see [Rez01, Section 6] and for a comparison with other models see [JT07, Ber10]. This section will serve as a short reminder on thecovariant model structure and all of its relevant definitions and theorems. For more details see [Ras17b],where all these definitions and theorems are discussed in more detail.A Reedy fibration p : L → X ( q : R → X ) is called a left fibration ( right fibration ) if the following is ahomotopy pullback square (using the notation Subsection 1.2(2)) L n X n L X < > ∗ p n p p < > ∗ , R n X n R X < n > ∗ q n p q < n > ∗ .Left fibrations (right fibrations) come with a model structure that has many desirable properties: There isunique left proper combinatorial simplicial model structure on the over category s S / X , called the covariantmodel structure ( contravariant model structure ). Here we will only state the relevant properties of the covariantmodel structure:(1) [Ras17b, Theorem 3.12] The fibrant object are left fibrations.(2) [Ras17b, Lemma 3.10] For a Reedy fibration p : Y → X , the following are equivalent NIMA RASEKH (I) p is a left fibration.(II) For every map σ : F ( n ) × ∆ [ l ] → X the induced map σ ∗ Y → F ( n ) × ∆ [ l ] is a left fibration.(III) For every map σ : F ( n ) → X the induced map σ ∗ Y → F ( n ) is a left fibration.(3) [Ras17b, Theorem 4.34] For a map of left fibrations g : L → M the following are equivalent:(I) g : L → M is a Reedy equivalence.(II) g : L → M is a Kan equivalence.(III) For every x : F ( ) → X , F ( ) × X Y → F ( ) × X Z is a diagonal equivalence of simplicial spaces.(4) [Ras17b, Theorem 4.39] A map f is a covariant equivalence if and only if for every map x : F ( ) → X , if the diagonal of the induced map Y × X R x → Z × X R x is a Kan equivalence. Here R x is the right fibrant replacement of the map x over X .(5) [Ras17b, Theorem 3.17] The following adjunction ( s S / X ) Cov ( s S / X ) Diagid ⊥ id is a Quillen adjunction, which is a Quillen equivalence if X is homotopically constant. Here theleft hand side has the covariant model structure and the right hand side has the induced diagonalmodel structure.(6) [Ras17b, Theorem 4.28] Let p : R → X be a right fibration. The following is a Quillen adjunction: ( s S / X ) Cov ( s S / X ) Covp ! p ∗ ⊥ p ∗ p ∗ .(7) [Ras17b, Lemma 3.25] Let i : A → B and j : C → D be cofibrations of simplicial spaces over X . If i or j are trivial cofibrations in the covariant model structure, then i (cid:3) j is a trivial cofibration as well.(8) [Ras17b, Theorem 5.1] Let f : X → Y be a map of simplicial spaces. Then the adjunction ( s S / X ) Cov ( s S / Y ) Covf ! ⊥ f ∗ is a Quillen adjunction, which is a Quillen equivalence whenever f is a CSS equivalence.(9) [Ras17b, Theorem 5.11] The following is a Quillen adjunction ( s S / X ) CSS ( s S / X ) Covid ⊥ id where the left hand side has the induced CSS model structure and the right hand side has thecovariant model structure.(10) [Ras17b, Theorem 4.18] For a small category C there are Quillen equivalencesFun ( C , S Kan ) proj ( s S / N C ) Cov
Fun ( C , S Kan ) projs R C s H C ⊥ s T C s I C ⊥ where the middle one has the covariant model structure the other ones have the projective modelstructure.(11) [Ras17b, Corollary 5.18] Let f : X → Y be a CSS equivalence and p : L → Y a left fibration over Y .Then the map f ∗ L → L is also a CSS equivalence.Left fibrations model maps into spaces. Our overall goal in this paper is it to generalize all aforemen-tioned results to the level of presheaves into higher categories. However, before we can do so we have toexpand our playing field, which leads us to the next section. ss S = Fun ( ∆ op , s S ) denotes the category of bisimplicial spaces (trisimplicialsets). We have the following basic notations with regard to bisimplicial spaces: ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 9 (1) Denote by F ( k , n ) the bisimplicial space defined as F ( k , n ) abc = Hom ∆ ([ a ] , [ k ]) × Hom ∆ ([ b ] , [ n ]) .Note in particular we have bijection F ( k , n ) ∼ = F ( k , 0 ) × F ( n ) .(2) Let ∂ F ( k , n ) → F ( k , n ) denote the map ( ∂ F ( k , 0 ) → F ( k , 0 )) (cid:3) ( ∂ F ( n ) → F ( n )) , which we con-sider the boundary of F ( k , n ) .(3) The category ss S is enriched over spacesMap ss S ( X , Y ) n = Hom ss S ( X × ∆ [ n ] , Y ) .(4) The category ss S is also enriched over itself ( Y X ) knl = Hom ss S ( X × F ( k , n ) × ∆ [ l ] , Y ) .(5) By the Yoneda lemma, for a simplicial space X we have a bijection of spaces X kn ∼ = Map s S ( F ( k , n ) , X ) . The category of bisimplicial spaces has a Reedymodel structure [Ree74], which is defined as follows:(F) A map f : Y → X is a (trivial) Reedy fibration if for each k , n ≥ ss S ( F ( k , n ) , Y ) → Map ss S ( ∂ F ( k , n ) , Y ) × Map ss S ( ∂ F ( k , n ) , X ) Map ss S ( F ( k , n ) , X ) is a (trivial) Kan fibration(W) A map f : Y → X is a Reedy equivalence if it is a level-wise Kan equivalence.(C) A map f : Y → X is a Reedy cofibration if it is an inclusion.The Reedy model structure is combinatorial , simplicial and proper . Moreover, it is also compatible with Cartesianclosure (Remark 1.1). These properties in particular imply that we can apply Bousfield localizations to theReedy model structure. See [Hir03] for more details. In order to avoid confusion we will call the Reedymodel structure on bisimplicial spaces, the biReedy model structure. In [Ras17b, Subsection 2.5] we studied important localizations ofthe Reedy model structure on simplicial spaces that are Quillen equivalent to the Kan model structure. Ina similar manner, we need localizations of the biReedy model structure that are Quillen equivalent to theReedy model structure, so we will introduce them right here. We will only state the relevant notation andleave the theorems without proofs.
Notation . Let Diag : ∆ × ∆ → ∆ × ∆ × ∆ be the functor given by Diag ([ n ] , [ l ]) = ([ n ] , [ l ] , [ l ]) . Similarly, for j =
1, 2 let p , p : ∆ × ∆ × ∆ → ∆ × ∆ be given by p ([ k ] , [ n ] , [ l ]) = ([ n ] , [ l ]) and p ([ k ] , [ n ] , [ l ]) = ([ k ] , [ l ]) .We want show that ss S has a model structure such that (( Diag ) ∗ , ( Diag ) ∗ ) becomes a Quillen equiva-lence. Theorem 1.8.
There is a unique, cofibrantly generated, simplicial model structure on ss S , called the diagonal ReedyModel Structure and denoted by ss S DiagRee , with the following specifications. C A map f : X → Y is a cofibration if it is an inclusion. W A map f : X → Y is a weak equivalence if ( Diag ) ∗ ( f ) : ( Diag ) ∗ ( X ) → ( Diag ) ∗ ( Y ) is a Reedyequivalence. F A map f : X → Y is a fibration if it satisfies the right lifting condition for trivial cofibrations.In particular, an object W is fibrant if it is biReedy fibrant and the map ( p ) ∗ ( p ) ∗ W → W is a biReedy equivalences.
These model structures all give us following long chain of Quillen equivalences.
Theorem 1.9.
The following is a simplicially enriched Quillen equivalencess S DiagRee s S Ree ( Diag ) ∗ ⊥ ( Diag ) ∗ . The proof is analogous to the proof of [Ras17b, Theorem 2.13].
Some functors we have defined until now will beparticularly important and hence we will give them more descriptive names. We use the following notationfor three functors ss S → s S : • LF ib n = ( p F ( n ) ) ∗ : The underlying n-level left fibration . In particular, if n = LF iband call it the underlying left fibration . • V al k = ( p F ( k ,0 ) ) ∗ : The k-level value . In particular, if k = V al and call it the value . • D iag = Diag : The diagonal .On the other hand, we use the following notation for two functors s S → ss S : • LE mb = ( p ) ∗ : The left fibration embedding . • VE mb = ( p ) ∗ : The value embedding .The terminology above is motivated by the fact that in the next section we will define a new notion offibration p : Y → X such that LF ib ( p ) : LF ib ( Y ) → X is a left fibration and V al ( Y ) will give us the fibers,representing the values . T HE R EEDY C OVARIANT M ODEL S TRUCTURE
In this section we generalize the covariant model structure to the category of bisimplicial spaces over asimplicial space. This gives us a good model for maps valued in simplicial spaces and the room we need tofurther define new model structures.
Notation . For the remaining sections we will fix a simplicial space X and denote the bisimplicial space LF ib ( X ) simply by X to simplify notation. Definition 2.2.
Let X be a simplicial space. We say a map of bisimplicial spaces p : Y → X is a Reedy leftfibration if it is a biReedy fibration and for all k , n ≥ Y kn Y k X n X < > ∗ p kn p p k < > ∗ .Notice this definition is equivalent to saying that the map is a biReedy fibration and for any k ≥ Y k → X is a left fibration. As in the case of left fibrations this construction comes with a model structure,the Reedy covariant model structure . Theorem 2.3.
Let X be a simplicial space considered a bisimplicial space via Notation 2.1. There is a unique simplicialcombinatorial left proper model structure on the category ss S / X , called the Reedy covariant model structure anddenoted by ( ss S / X ) ReeCov , which satisfies following conditions:(1) An object L → X is fibrant if it is a Reedy left fibration.
ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 11 (2) A map is a cofibration if it is a monomorphism.(3) A map is a weak equivalence if it is a level-wise covariant equivalence over X.(4) A weak equivalence (fibration) between Reedy left fibrations is precisely level-wise Reedy equivalence (biReedyfibration).Proof.
Starting with the Reedy model structure on ( s S / X ) Ree , we can construct two model structures on thecategory ss S / X .(1) First, we can localize the Reedy model structure with respect to map F ( ) → F ( n ) → X to get thecovariant model structure ( s S / X ) Cov . Then we can take simplicial objects in this model structure,which gives us the category ss S / X , and give it the Reedy model structure.It immediately satisfies following conditions: • It is simplicial combinatorial left proper. • Cofibrations are monomorphisms. • Weak equivalences are level-wise covariant equivalences over X .(2) Alternatively, we can first take the Reedy model structure on ss S / X . Then we can localize the modelstructure with respect to maps F ( k , 0 ) → F ( k , n ) → X to get a model structure on ss S / X whichimmediately satisfies following conditions: • It is simplicial combinatorial left proper. • Cofibrations are monomorphisms • The fibrant objects are Reedy left fibrations. • A weak equivalence (fibration) between Reedy left fibrations is precisely level-wise Reedyequivalence (biReedy fibration).Thus, the theorem follows if we can prove that these two model structures coincide. As we already havethe same cofibrations, it suffices to prove both model structures have the same fibrant objects. In order todo that we need to better understand the fibrant objects in the two model structures.Let p : L → X be a map of bisimplicial spaces. Let M k L be the corresponding matching object. An object,is fibrant in the first model structure if the maps L k → M k L are left fibrations of simplicial spaces for all k ≥
0, where M k L is the matching object [Hov99, Section 5.2]. On the other side, p is fibrant in the secondmodel structure if the maps L k → X are left fibrations of simplicial spaces for all k ≥
0. We need to provethese two conditions coincide.In order to prove it we use following commutative triangle L k M k LX .Let p be fibrant in the first model structure. We want to prove that L k → X is a left fibration. We proceedby induction. The case k = M L = X . Assume that L , ..., L k are left fibrationsover X . We want to prove that L k + → X is a left fibration over X . By construction the map M k + L → X isa limit of a diagram in s S / X with value objects L m → X (where m ≤ k ). By induction assumption these areall left fibrations and left fibrations are closed under limits and so M k + L → X is a left fibration. The resultnow follows from the fact that left fibrations are closed under composition.On the other side assume p is fibrant in the second model structure. We want to prove that L k → M k → L is a left fibration. By assumption L k → X are left fibrations for all k and so M k L → X is also a left fibration,as it is a limit with value L k . The result now follows from the fact that in the commutative triangle abovethe two legs are left fibrations. (cid:3) The key input of the proof is that the following two different ways of constructing model structures on ss S / X coincide: ( s S / X ) Ree ( ss S / X ) Ree ( s S / X ) Cov ( ss S / X ) ReeCovReecov covRee .So, the Reedy covariant model structure on bisimplicial spaces over X has two perspectives: • It is a Reedy model structure, which allows us to easily characterize the weak equivalences. • It is a localization model structure, which allows us to easily characterize the fibrant objects.That is the reason why we were able to give such an elegant characterization of the Reedy covariant modelstructure. We can now use this characterization to directly generalize results about left fibrations to Reedyleft fibrations, using the fact that many results about a model category generalize to its Reedy model cate-gory.
Remark . In analogy with the duality between left and right fibrations, we also have a notion of
Reedyright fibrations and similarly, a
Reedy contravariant model structure , which can be defined and constructedsimilar to Theorem 2.3. Hence we will refrain from making those similar definitions explicit.
Lemma 2.5.
Let p : Y → X be a biReedy fibration over X. The following are equivalent.(1) p is a Reedy left fibration.(2) For every map σ : F ( n ) × ∆ [ l ] → X the induced map σ ∗ Y → F ( n ) × ∆ [ l ] is a Reedy left fibration.(3) For every map σ : F ( n ) → X the induced map σ ∗ Y → F ( n ) is a Reedy left fibration. This follows from the fact that Reedy left fibrations are determined level-wise and Subsection 1.6(2).
Theorem 2.6.
Let f : X → Y be map of simplicial spaces. Then the following adjunction ( ss S / X ) ReeCov ( ss S / Y ) ReeCovf ! ⊥ f ∗ is a Quillen adjunction, which is a Quillen equivalence if f is a CSS equivalence. This follows directly from applying [Hir03, Proposition 15.4.1] to Subsection 1.6(8). Similarly applyingthe same proposition to Subsection 1.6(5) gives us the next result.
Theorem 2.7.
The following adjunction ( ss S / X ) ReeCov ( ss S / X ) DiagReeid ⊥ id is a Quillen adjunction, which is a Quillen equivalence if X is homotopically constant. Here the left hand side hasthe Reedy covariant model structure and the right hand side has the induced diagonal Reedy model structure. Inparticular, the diagonal Reedy model structure is a localization of the Reedy covariant model structure. Lemma 2.8.
Let i : A → B and j : C → D be cofibrations of bisimplicial spaces over X. If i or j are trivialcofibrations in the Reedy covariant model structure, then i (cid:3) j is a trivial cofibration as well.
This follows from Subsection 1.6(7) and the fact that cofibrations and trivial cofibrations are determinedlevel-wise.
Theorem 2.9.
Let p : R → X be a Reedy right fibration. The following is a Quillen adjunction ( ss S / X ) ReeCov ( ss S / X ) ReeCovp ! p ∗ ⊥ p ∗ p ∗ . ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 13
This follows from applying [Hir03, Proposition 15.4.1] to Subsection 1.6(6).
Theorem 2.10.
Let X be a simplicial space considered a bisimplicial space (Notation 2.1). Then a map of bisimplicialspaces Y → Z over X is a Reedy covariant equivalence if and only if for each map x : F ( ) → X the induced mapY × X LF ib ( R x ) → Z × X LF ib ( R x ) is a diagonal Reedy equivalence. Here R x is a choice of contravariant fibrant replacement of x in s S / X . This follows from Subsection 1.6(4) and the fact that weak equivalences are determined level-wise.
Remark . It is interesting to compare this result to the one for simplicial spaces. Despite the fact that wegeneralized everything to the bisimplicial setting, the contravariant fibrant replacements have remainedsimplicial spaces.The underlying reason is that for a map x : F ( ) → X , contravariant fibrant replacements and Reedycontravariant fibrant replacements are the same. Indeed for an arbitrary Reedy right fibration R → X wehave Map ss S / X ( F (
0, 0 ) , R ) ≃ −→ Map s S / X ( F ( ) , R ) ≃ −→ Map s S / X ( R x , R ) ≃ −→ Map ss S / X ( R x , R ) where we used the fact that R → X is a right fibration.Similar to the case of covariant model structure, weak equivalences between fibrant objects can be char-acterized in much easier ways applying Subsection 1.6(3) level-wise. Theorem 2.12.
Let L and M be two Reedy left fibrations over X. Let g : L → M be a map over X. Then the followingare equivalent.(1) g : L → M is a biReedy equivalence.(2) V al ( g ) : V al ( Y ) → V al ( Z ) is a Reedy equivalence.(3) For every x : F ( ) → X, F ( ) × X Y → F ( ) × X Z is a diagonal Reedy equivalence of bisimplicial spaces.
Finally, we can also recover the
Grothendieck construction . Let ss R C , ss H C , ss T C and ss I C be the functors s R C , s H C , s T C and s I C defined level-wise, as further explained in Subsection 1.6(10). Applying [Hir03,Proposition 15.4.1] to these adjunctions gives us following result. Theorem 2.13.
Let C be a small category. The two simplicially enriched adjunctions Fun ( C , s S Ree ) proj ( ss S / N C ) ReeCov
Fun ( C , s S Ree ) projss R C ss H C ⊥ ss T C ss I C ⊥ are Quillen equivalences. Here Fun ( C , s S ) has the projective model structure and ss S / N C has the Reedy covariantmodel structure over N C .Remark . Here we only mentioned the Grothendieck construction over nerves of categories. How-ever, we also have a Grothendieck construction over arbitrary simplicial spaces. Indeed, this follows fromthe Quillen equivalence between the covariant model structure over simplicial spaces and simplicial sets([Ras17b, Appendix B]) and the straightening construction for the covariant model structure [Lur09a, Chap-ter 2]. L
OCALIZATIONS OF R EEDY L EFT F IBRATIONS
In Section 2 we defined fibrations which we should think of as modeling functors valued in simplicialspaces (as has been illustrated in Theorem 2.13). In this section we want to study functors valued in local-izations of simplicial spaces. In the next section we will then apply these results to functors valued in Segalspaces, complete Segal spaces and homotopically constant simplicial spaces.
Notation . As this whole section is focused on the study of Bousfield localizations we will establishfollowing terminology with regard to localizations. • Throughout S will be a set of monomorphisms in the category simplicial spaces s S . • A simplicial space X is called local with respect to S if for every every map f : A → B in S ,Map s S ( B , X ) → Map s S ( A , X ) is a Kan equivalence. • A bisimplicial space X is called local with respect to S if V al ( X ) is local with respect to S . This isequivalent to Map ss S ( VE mb ( B ) , X ) → Map ss S ( VE mb ( A ) , X ) being a Kan equivalence for every A → B in S . • A map of bisimplicial spaces p : Y → X is called local with respect to S if the mapMap ss S ( VE mb ( B ) , Y ) → Map ss S ( VE mb ( A ) , Y ) × Map ss S ( VE mb ( A ) , X ) Map ss S ( VE mb ( B ) , X ) is a weak equivalence for every map f : A → B in S . Note this is equivalent to the condition thatfor every map f : A → B in S and every map VE mb ( B ) → X , the induced mapMap / X ( VE mb ( B ) , Y ) → Map / X ( VE mb ( A ) , Y ) is a Kan equivalence.We can now use the intuition outlined above to give following definition. Here, recall, for a given sim-plicial space X , we denote the bisimplicial space LE mb ( X ) again by X , to simplify notation (Notation 2.1). Definition 3.2.
Let X be a simplicial space and S a set of monomorphisms of simplicial spaces (not over X ).Then a Reedy left fibration p : L → X is an S-localized Reedy left fibration if it is local with respect to S .The goal of this section is to study this fibration. In particular, we want to show:(1) It comes as the fibrant objects of a model structure on ss S / X (Theorem 3.5).(2) We can give various alternative characterizations of the fibrant objects (Theorem 3.15/Corollary 3.16).(3) We can give a detailed characterization of the weak equivalences (Theorem 3.20). In this subsection we want prove that S -localizedReedy left fibrations are fibrant objects in a model structure, the S-localized Reedy covariant model structure .Moreover, in order to study its fibrant objects and weak equivalences (in Subsection 3.2), we introduceseveral related model structures, the
S-localized Reedy model structure and diagonal S-localized Reedy modelstructure . Finally, we end this subsection by proving a Grothendieck construction for S -localized Reedy leftfibrations over nerves of categories. Theorem 3.3.
Let S be a set of monomorphisms of simplicial spaces. There is a unique, combinatorial left propersimplicial model structure on s S , denoted by s S Ree S and called the S-localized Reedy model structure, defined asfollows. C A map Y → Z is a cofibration if it is an inclusion. F An object W is fibrant if it is Reedy fibrant and local with respect to S. W A map Y → Z is a weak equivalence if for every fibrant object W the map
Map s S ( Z , W ) → Map s S ( Y , W ) is a Kan equivalence.Proof. This is a direct application of a left Bousfield localization to the Reedy model structure on simplicialspaces [Hir03, Theorem 4.1.1]. (cid:3)
ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 15
Theorem 3.4.
Let S be a set of monomorphisms of simplicial spaces. There is a unique simplicial combinatorial leftproper model structure on ss S , denoted by ss S DiagRee S and called the diagonal S-localized Reedy model structure,defined as follows.(1) A map Y → Z is a cofibration if it is an inclusion.(2) A map g : Y → Z is a weak equivalence if the diagonal map D iag ( g ) : D iag ( Y ) → D iag ( Z ) is an S-localized Reedy equivalence.(3) An object W is fibrant if and only if it is fibrant in the diagonal Reedy model structure and local with respectto S.(4) The following adjunction ( ss S ) DiagRee S s S Ree S D iag =( Diag ) ∗ ⊥ ( Diag ) ∗ is a Quillen equivalence. Here the left hand side has the diagonal localized Reedy model structure and theright hand side has the localized Reedy model structure.Proof. By Theorem 1.9 we have a simplicial Quillen equivalence ( ss S ) DiagRee s S Ree D iag =( Diag ) ∗ ⊥ ( Diag ) ∗ which gives us a simplicial Quillen adjunction with fully faithful derived right adjoint ( ss S ) DiagRee s S Ree S D iag =( Diag ) ∗ ⊥ ( Diag ) ∗ .Hence, by [Lur09a, Corollary A.3.7.10] there exists a new unique simplicial, combinatorial, left propermodel structure on ss S , the diagonal S-localized model structure , which satisfies following conditions:(1) Cofibrations are monomorphisms.(2) Weak equivalences are S -localized diagonal equivalences.(3) The adjunction ( D iag = ( Diag ) ∗ , ( Diag ) ∗ ) is a simplicial Quillen equivalence between this modelstructure and the S -localized Reedy model structure on s S .(4) An object X is fibrant if it is biReedy fibrant and biReedy equivalent to ( Diag ) ∗ ( Y ) where Y is S -local, meaning that X is fibrant in the diagonal Reedy model structure and S -local. (cid:3) Theorem 3.5.
Let S be a set of monomorphisms of simplicial spaces. There is a unique simplicial combinatorialleft proper model structure on ss S / X , denoted by ( ss S / X ) ReeCov S and called the S-localized Reedy covariant modelstructure, defined as follows. C A map Y → Z over X is a cofibration if it is an inclusion. F An object Y → X is fibrant if it is a Reedy left fibration and local with respect to S. W A map Y → Z over X is a weak equivalence if for every fibrant object W → X the map
Map / X ( Z , W ) → Map / X ( Y , W ) is a Kan equivalence.Proof. Notice the model structure on the category ss S / X is still proper and cellular [Hir03, Proposition12.1.6] and so we can apply left Bousfield localization [Hir03, Theorem 4.1.1] with respect to the set ofmorphisms L = { VE mb ( A ) → VE mb ( B ) → X : A → B ∈ S } . (cid:3) Combining Theorem 2.7 with Theorem 1.2 gives us following similar result.
Proposition 3.6.
The following adjunction ( ss S / X ) ReeCov S ( ss S / X ) DiagRee S id ⊥ id is a Quillen adjunction, which is a Quillen equivalence whenever X is homotopically constant. Here the left handside has the localized Reedy covariant model structure and the left hand side has the induced diagonal localized Reedymodel structure over the base X. One very important instance is the case X = F ( ) . The theorem shows that ss S ReeCov S is the same as ss S DiagRee S .We move on to prove the Grothendieck construction for localized Reedy left fibrations over nerves ofcategories. Before that let us recall that an object in the projective model structure on Fun ( C , s S Ree S ) proj (where s S has the S -localized Reedy model structure) is fibrant if it is fibrant in the projective model struc-ture on Fun ( C , s S Ree ) proj (where now s S has the Reedy model structure) and is local with respect to naturaltransformations id × f : Hom C ( c , − ) × A → Hom C ( c , − ) × B for all objects c and maps f : A → B in S . Theorem 3.7.
Let C be a small category. Then the adjunctions defined in Theorem 2.13 Fun ( C , s S Ree S ) proj ( ss S / N C ) ReeCov S Fun ( C , s S Ree S ) projss R C ss H C ⊥ ss T C ss I C ⊥ are simplicial Quillen equivalences. Here the middle has the S-localized Reedy covariant model structure and the twosides have the projective model structure on the S-localized Reedy model structure.Proof. First we show both are Quillen adjunctions. As both left adjoints still preserve cofibrations by[Lur09a, Corollary A.3.7.2] it suffices to prove that the right adjoints preserve fibrant objects. By Theorem 2.13,the right adjoints preserve fibrant objects in the unlocalized model structures, so we only need to confirmthat they preserve local objects.Using the adjunctions this is equivalent to proving the following statements: ss Z C Hom C ( c , − ) × A → ss Z C Hom C ( c , − ) × B is an S -localized Reedy covariant equivalence over C and ss T C ( VE mb ( A )) → ss T C ( VE mb ( B )) is an S -localized projective equivalence. Both of those are immediate computations.We now move on to proving they are Quillen equivalences. The composition map ss T C ◦ ss R C is natu-rally equivalent to the identity functor and so is a Quillen equivalence. Hence it suffices to prove that theadjunction ( s R C , ss H C ) is a Quillen equivalence.By Theorem 2.13, the counit map is an equivalence. So, we move on to the derived unit map. Again, byTheorem 2.13, for a fibrant object F : C → s S , the map ss R C F → ss I C F is a biReedy equivalence and hencea localized Reedy covariant equivalence. Hence ss I C F is a fibrant replacement of ss R C F . Thus, the derivedunit map is given by F → ss H C ss I C F ,which is indeed an equivalence as it is naturally equivalent to the identity as explained above. (cid:3) ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 17
The result has several important corollaries that we will use in the next subsection.
Corollary 3.8.
Let p : L → N C be a Reedy left fibration. Then p is an S-localized Reedy left fibration if and only ifit is fiberwise diagonal S-localized Reedy fibrant. Corollary 3.9.
A map of bisimplicial spaces Y → Z over N C is S-localized Reedy covariant equivalence if and onlyif the map Y × N C N C / c → Z × N C N C / c is a diagonal S-localized Reedy equivalence. If we let C = [ n ] , then N [ n ] = F ( n ) and so we can use the results above immediately to understandthe S -localized Reedy covariant model structure over F ( n ) . For many applications, however, this is notgood enough. We want to understand f -localized Reedy left fibrations over F ( n ) × ∆ [ l ] . For that we havefollowing result: Corollary 3.10.
The Reedy equivalence π : F ( n ) × ∆ [ l ] → F ( n ) induces a Quillen equivalence ( ss S / F ( n ) × ∆ [ l ] ) ReeCov S ( ss S / F ( n ) ) ReeCov S ( π ) ! ⊥ ( π ) ∗ and so, in particular, every S-localized Reedy left fibration is biReedy equivalent to a map of the form p × ∆ [ l ] : ss R C G × ∆ [ l ] → F ( n ) × ∆ [ l ] , where G : C → s S is a fibrant object in the projective model structure. We will use the local results in the next subsection to study S -localized Reedy left fibrations and theirequivalences over arbitrary simplicial spaces. In this subsection we want to studythe fibrant objects and weak equivalences in the S -localized Reedy covariant model structure over an arbi-trary simplicial space X .Some results will require some conditions on the set of maps S , which we will fix now. Notation . Let S be a set of monomorphisms of simplicial spaces. • (S) : A map f in S satisfies condition (S) if every homotopically constant simplicial space is local withrespect to f , which is equivalent to f being an equivalence in the diagonal model structure. The setof maps S satisfies condition (S) if every map in S satisfies condition (S) . • (D) : A map f : A → B in S satisfies condition (D) if B is diagonally contractible. The set of maps Ssatisfies condition (D) if every map in S satisfies condition (D) . • (C) : A map f : A → B satisfies condition (C) if it satisfies condition (S) and (D) . This an be stateddirectly as A and B being diagonally contractible. The set of maps S satisfies condition (C) if everymap in S satisfies condition (C) . • (P) : The set of monomorphisms S satisfies condition (P) if W being local with respect to S implies that W X is local with respect to S for all simplicial spaces X . Example 3.12.
Let us see some examples of maps that satisfy these conditions:(1) The simplicial space F ( n ) is a diagonally contractible. Hence any map A → F ( n ) satisfies condition (D) .(2) The simplicial space G ( n ) [Rez01, Section 5] is also diagonally contractible. Hence the inclusions G ( n ) → F ( n ) satisfy condition (C) .(3) The map also satisfies condition (P) [Rez01, Lemma 10.3].(4) Let C be a contractible category. Then any map F ( ) → N C satisfies condition (C) .(5) In particular, the map F ( ) → E ( ) (Subsection 1.3(4)) satisfies condition (C) , but also condition (P) [Rez01, Proposition 12.1]. We will start with characterizations of S -localized Reedy left fibrations. First two lemmas. Lemma 3.13.
Let p : L → X be a biReedy fibration. Then the following are equivalent:(1) For every map σ : F ( n ) × ∆ [ l ] → X, the pullback map σ ∗ p : σ ∗ L → F ( n ) × ∆ [ l ] is an S-localized Reedyleft fibration.(2) For every map σ : F ( n ) → X, the pullback map σ ∗ p : σ ∗ L → F ( n ) is an S-localized Reedy left fibration.(3) p is a Reedy left fibration and for every point { x } : F ( ) → X the fiber F ib x L is fibrant in the diagonalS-localized Reedy model structure.(4) p is a Reedy left fibration and for every point { x } : F ( ) → X the fiber V al ( F ib x L ) is fibrant in theS-localized Reedy model structure.Proof. All four statements break down into two parts: proving p is a Reedy left fibration and proving it islocal with respect to S . The first always follows either by definition or from Lemma 2.5. Hence we will onlyfocus on proving it is local with respect to S . (1) ⇔ (2) This follows from the fact that π : F ( n ) × ∆ [ l ] → F ( n ) is a Reedy equivalence and being localwith respect to S is invariant under Reedy equivalences. (2) ⇔ (3) One side is immediate, for the other we will use Theorem 3.7. Fix a map σ : F ( n ) → X . Wewant to prove σ ∗ p : σ ∗ L → F ( n ) is an S -localized Reedy left fibration. By assumption p : L → X is alreadya Reedy left fibration, which, by Lemma 2.5, implies that σ ∗ p is also a Reedy left fibration.Hence, by Theorem 2.13, σ ∗ p : σ ∗ L → F ( n ) is Reedy equivalent to a map ss R [ n ] G → F ( n ) , where G : [ n ] → s S and as the property of being local with respect to S is invariant under Reedy equivalences, σ ∗ p is an S -localized Reedy left fibration if and only if ss R [ n ] G is local with respect to S . By Theorem 3.7, thisitself is equivalent to G being fibrant in the projective model structure, which by definition means that forall 0 ≤ i ≤ n , G ( i ) is fibrant in the S -localized Reedy model structure. This is directly equivalent to σ ∗ p being fiber-wise fibrant in the diagonal S -localized Reedy model structure. (3) ⇔ (4) This follows from the definition of fibrant objects in the diagonal S -localized model structureon ss S . (cid:3) Lemma 3.14.
Assume that S satisfies condition (S) and p : L → S is a biReedy fibration. Then the following areequivalent.(1) p is an S-localized Reedy left fibration.(2) p is a Reedy left fibration and the simplicial space V al ( L ) is local with respect to S.(3) p is a Reedy left fibration and the simplicial spaces V al k ( L ) are local with respect to S for all k ≥ .Proof. (1) ⇔ (2) Let p be an S -localized Reedy left fibration. We have a commutative diagramMap / X ( VE mb ( B ) , L ) Map / X ( VE mb ( A ) , L ) Map / V al ( X ) ( B , V al ( L )) Map / V al ( X ) ( A , V al ( L )) ∼ = ∼ = .The vertical maps are bijections using the enriched adjunction ( VE mb, V al ) . So the top map is an equiva-lence (which is the definition of a localized Reedy left fibration) if and only if the bottom map is an equiva-lence (which is equivalent to V al ( L ) → V al ( X ) being fibrant in the localized model structure on s S / V al ( X ) ).As S satisfies condition (S) , V al ( X ) = X is local with respect to S and so the bottom map being anequivalence is equivalent to V al ( L ) being local with respect to S , finishing the proof. ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 19 (2) ⇔ (3) One side side is a special case. For the other side, notice we have a Reedy equivalence ofsimplicial spaces V al n ( L ) ≃ V al ( L ) × X n X .The right hand side is local with respect to S ( V al ( L ) by assumption and X n , X by condition (S) ), hence theright hand is local as well. (cid:3) Theorem 3.15.
If p is an S-localized Reedy left fibration, then it satisfies the conditions of Lemma 3.13. The oppositeholds if S satisfies condition (D) .Proof. If p is an S -localized Reedy left fibration, then it satisfies Condition ( ) of Lemma 3.13, as fibrationsare closed under pullback. On the other side, assume S satisfies condition (D) and assume p is a biReedyfibration. We will prove that Condition ( ) of Lemma 3.13 implies p is an S -localized Reedy left fibration.By Lemma 2.5, Condition (2) implies that p is a Reedy left fibration, so we only need to show it is localwith respect to S . It suffices to prove that p satisfies the right lifting property with respect to the cofibration j defined as the pushout product j = ( VE mb ( f ) : VE mb ( A ) → VE mb ( B )) (cid:3) ( ∂ ∆ [ n ] → ∆ [ n ]) ,where f is in S . The codomain of j is VE mb ( B ) × ∆ [ n ] and every map σ : VE mb ( B ) × ∆ [ n ] → X factorsthrough a map δ : ∆ [ n ] → X (as S satisfies condition (D) . Hence we get following diagram VE mb ( A ) × ∆ [ n ] ∐ VE mb ( A ) × ∂ ∆ [ n ] VE mb ( B ) × ∂ ∆ [ n ] δ ∗ L L VE mb ( B ) × ∆ [ n ] ∆ [ n ] X j δ ∗ p p σ δ .By assumption δ ∗ L → ∆ [ n ] is an S -localized Reedy left fibration and so has a lift, which implies that ouroriginal lifting problem has a solution proving that p is an S -localized Reedy left fibration. (cid:3) Combining Theorem 3.15 with Lemma 3.14 immediately gives us following result.
Corollary 3.16.
Let S satisfy condition (C) . Then all conditions in Lemma 3.14 and Lemma 3.13 coincide.
We move on to characterize weak equivalences in the S -localized Reedy covariant model structure. First,observe that we have a very immediate result for weak equivalences between fibrant objects. Theorem 3.17.
Let L and M be two S-localized Reedy left fibrations over X. Let g : L → M be a map over X. Thenthe following are equivalent.(1) g : L → M is a biReedy equivalence.(2) V al ( g ) : V al ( L ) → V al ( M ) is a Reedy equivalence.(3) For every { x } : F ( ) → X, the map F ib x V al ( L ) → F ib x V al ( M ) is a Reedy equivalence of bisimplicialspaces.(4) For every { x } : F ( ) → X, the map F ib x ( L ) → F ib x ( M ) is a diagonal Reedy equivalence of bisimplicialspaces. Before moving to the general case, we prove a recognition principle for S-localized Reedy covariant equiva-lences between Reedy left fibrations . For the next proposition we need following construction. Let p : L → X be a Reedy left fibration. Thenwe have following diagram(3.18) L • ˜ L • ˆ LX pi ≃ ˜ p j ≃ ˆ p .Here the first map is the level-wise functorial factorization of the simplicial object in ( s S / X ) Ree S in the S -localized Reedy model structure. Moreover, let ˆ p : ˆ L → X be the biReedy fibrant replacement over X . Proposition 3.19.
Let S satisfy condition (C) . Let p : L → X, q : M → X be Reedy left fibrations (not necessarilylocalized) and let f : L → M be a map over X. Then the following are equivalent:(1) f is an S-localized Reedy covariant equivalence.(2) The map ˆ f : ˆ L → ˆ M constructed in 3.18 is a biReedy equivalence.(3) The map V al ( ˆ f ) : V al ( ˆ L ) → V al ( ˆ M ) constructed in 3.18 is a Reedy equivalence.(4) The map V al ( f ) : V al ( L ) → V al ( M ) is an S-localized Reedy equivalence.(5) For every object { x } : F ( ) → X, the induced map on fibers V al ( F ib x L ) → V al ( F ib x M ) is an S-localized Reedy equivalence.(6) For every object { x } : F ( ) → X, the induced map on fibers F ib x L → F ib x Mis a diagonal S-localized Reedy equivalence.Proof. (1) ⇔ (2) It suffices to prove that the map ˆ p : ˆ L → X from 3.18 is a fibrant replacement of L → X inthe S -localized Reedy covariant model structure.For that we need to prove two statements: • ˆ p : ˆ L → X is an S -localized Reedy left fibration: Indeed it is biReedy fibrant by definition. Moreover, V al ( ˆ L ) is Reedy equivalent to V al ( ˜ L ) , which is by definition fibrant in the S -localized Reedy modelstructure, and so is itself fibrant in the S -localized Reedy model structure. Finally, ˆ L → X is Reedyleft fibration, as it is biReedy equivalent to ˜ L → X and for every n ≥ V al n ( L ) V al n ( ˆ L ) V al ( L ) × X X n V al ( ˆ L ) × X X n V al n ( i ) ≃≃ V al ( i ) × X X n ≃ . • j ◦ i is an S -localized Reedy covariant equivalence. Indeed i is a level-wise S -localized Reedy equiv-alence and so an equivalence in the S -localized Reedy covariant model structure and j is a biReedyequivalence.Now that we have established that ˆ L is the S -localized Reedy covariant fibrant replacement of L over X ,it follows by definition of Bousfield localizations that f is an S -localized Reedy covariant equivalence if andonly if ˆ f is a biReedy equivalence. ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 21 (2) ⇔ (3) ˆ L and ˆ M are Reedy left fibrations and so, by Theorem 2.12, a map ˆ f : ˆ L → ˆ M is a biReedyequivalence if and only if V al ( ˆ f ) : V al ( ˆ L ) → V al ( ˆ M ) is a Reedy equivalence. (3) ⇔ (4) We have a commutative diagram V al ( L ) V al ( ˆ L ) V al ( M ) V al ( ˆ M ) V al ( f ) V al ( ˆ f ) .By construction, the horizontal maps are fibrant replacements in the S -localized Reedy model structure.Hence, V al ( f ) is an S -localized Reedy weak equivalence if an only if V al ( ˆ f ) is a Reedy equivalence. (3) ⇔ (5) First, observe that V al ( ˆ L ) → V al ( ˆ M ) is a Reedy equivalence if and only if for every { x } : ∆ [ ] → X , the induced map F ib x ( V al ( ˆ L )) → F ib x ( V al ( ˆ M )) is a Reedy equivalence.Now, for a given point { x } : ∆ [ ] → X . The induced map on fibers F ib x ( V al ( L )) → F ib x ( V al ( ˆ L )) is still the fibrant replacement in the S -localized Reedy model structure. Hence this is equivalent to F ib x ( V al ( L )) → F ib x ( V al ( M )) being a S -localized Reedy equivalence. (5) ⇔ (6) This follows from the fact that L → X is a Reedy left fibration and so VE mb V al F ib x ( L ) → F ib x ( L ) is a biReedy equivalence. (cid:3) Theorem 3.20.
A map g : Y → Z of bisimplicial spaces over X is an equivalence in the localized Reedy covariantmodel structure if and only if for each map { x } : F ( ) → X, the induced mapY × X LF ib ( R x ) → Z × X LF ib ( R x ) is an equivalence in the diagonal localized Reedy model structure. Here R x is a choice of right fibrant replacement ofthe map { x } .Proof. Let ˆ g : ˆ Y → ˆ Z be a fibrant replacement of g in the Reedy covariant model structure (note: notlocalized). Moreover, let { x } : F ( ) → X be a vertex in X . This gives us following zig-zag of maps:ˆ Y × X F ( ) ˆ Z × X F ( ) ˆ Y × X R x ˆ Z × X R x Y × X R x Z × X R xReeContra ≃ ReeContra ≃ ReeCov ≃ ReeCov ≃ .According to Theorem 2.9 the top vertical maps are Reedy contravariant equivalences and the bottom verti-cal maps are Reedy covariant equivalences. By Theorem 2.7 both of these are diagonal Reedy equivalences,which are always diagonal localized Reedy equivalences (Theorem 3.4). Thus the top map is a diagonallocalized Reedy equivalence if and only if the bottom map is one, but by Proposition 3.19 this is equivalentto Y → Z being a localized Reedy contravariant equivalence over X . (cid:3) Theorem 3.21.
Let g : X → Y be a map of simplicial spaces. Then the adjunction ( ss S / X ) ReeCov S ( ss S / Y ) ReeCov S g ! ⊥ g ∗ is a Quillen adjunction, which is a Quillen equivalence whenever g is a CSS equivalence. Here both sides have theS-localized Reedy covariant model structure.Proof. Clearly it is a Quillen adjunction as fibrations are stable under pullback.Let us now assume that g is a CSS equivalence. We want to prove that ( g ! , g ∗ ) is a Quillen equivalence of S -localized Reedy covariant model structures. By Theorem 2.6 it is a Quillen equivalence of Reedy covariantmodel structures and we want to use Theorem 1.2 to finish the proof. Unfortunately we cannot applyit directly as we have not characterized the S -localized Reedy covariant model structure on ss S / Y via g ! .Hence, we will prove that in this case they coincide.We need to prove the following fact: Let p : L → Y be a Reedy left fibration. Then p is S -localized ifand only and only if g ∗ p : g ∗ L → X is an S -localized Reedy left fibration. By Subsection 1.6(11), g ∗ L → L is a level-wise CSS equivalence, which means it is a level-wise covariant equivalence (Subsection 1.6(9)).Hence, if g ∗ p is S -localized then p is S -localized as well. (cid:3) Theorem 3.22.
Let S satisfy conditions (P) and (C) . Let p : R → X be a Reedy right fibration over X. The inducedadjunction ( ss S / X ) ReeCov S ( ss S / X ) ReeCov S p ! p ∗ ⊥ p ∗ p ∗ is a simplicial Quillen adjunction. Here both sides have the S-localized Reedy covariant model structure.Proof. Clearly the left adjoint preserves cofibrations and so by [Lur09a, Corollary A.3.7.2] it suffices to showthat the right adjoint preserves fibrant objects. So, let L → X be a localized Reedy left fibration over X .Then we have to show that p ∗ p ∗ L → X is also a localized Reedy left fibration over X . By Theorem 2.9, wealready know that it is a Reedy left fibration, so all that is left is to show that it is local with respect to S . ByDefinition 3.2, it suffices to show that for any map q : VE mb ( B ) → X the induced mapMap / X ( VE mb ( B ) , p ∗ p ∗ L ) → Map / X ( VE mb ( A ) , p ∗ p ∗ L ) is a Kan equivalence. Using the adjunction, this is equivalent toMap / X ( p ! p ∗ VE mb ( B ) , L ) → Map / X ( p ! p ∗ VE mb ( A ) , L ) being a Kan equivalence. For that it suffices to show that p ! p ∗ VE mb ( A ) → p ! p ∗ VE mb ( B ) is a localized Reedy covariant equivalence over X .As S satisfies condition (C) , D iag ( B ) is contractible and so the map VE mb ( B ) → X is Reedy equivalentto a map of the form VE mb ( B ) → F ( ) → X . Thus p ∗ ( VE mb ( B )) = VE mb ( B ) × X R ≃ VE mb ( B ) × ( F ( ) × X R ) similarly p ∗ ( VE mb ( A )) ≃ VE mb ( A ) × ( F ( ) × X R ) . However, VE mb ( A ) × ( F ( ) × X R ) → VE mb ( B ) × ( F ( ) × X R ) is a localized Reedy covariant equivalence over X . Indeed, this immediately follows from the fact that S satisfies condition (P) . (cid:3) Using condition (P) we can recover other interesting results about S -localized Reedy left fibrations. ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 23
Proposition 3.23.
Let S be a set of cofibrations that satisfy condition (P) . Let g : C → D be a cofibration ofbisimplicial spaces and p : L → X a S-localized Reedy left fibration. Then exp ( g , p ) is also a localized Reedy leftfibration.Proof. By Subsection 1.6(6) exp ( g , p ) is a Reedy left fibration and so it suffices to prove that it is local withrespect to S . It suffices to observe that exp ( f , exp ( g , p )) is a trivial biReedy fibration for every f in S . Bydirect computation we haveexp ( f , exp ( g , p )) ∼ = exp ( f (cid:3) g , p ) ∼ = exp ( g , exp ( f , p )) .The result now follows from the fact that exp ( f , p ) is a trivial biReedy fibration (as f satisfies (P) ) and thebiReedy model structure is compatible with Cartesian closure (Subsection 1.8). (cid:3) Corollary 3.24.
Let S be a set of cofibrations that satisfy condition (P) . Let L → X be an S-localized Reedy leftfibration. Then for any bisimplicial space Y, L Y → X Y is also an S-localized Reedy left fibration. (S EGAL ) C
ARTESIAN F IBRATIONS
In the previous we section defined and studied fibrations with fiber localizations of Reedy fibrant sim-plicial spaces. In this section we want to apply these results to three very important cases: Segal spaces,complete Segal spaces and homotopically constant simplicial spaces. Similar to the previous sections X is afixed simplicial space considered a bisimplicial space (Notation 2.1). Also recall the notion of being local asdescribed in Notation 3.1. Definition 4.1.
We say a Reedy left (right) fibration Y → X over X is a Segal coCartesian fibration ( SegalCartesian fibration ) if it is local with respect to the set of maps S = { G ( n ) → F ( n ) : n ≥ } . Definition 4.2.
We say a Reedy left (right) fibration Y → X is a coCartesian fibration ( Cartesian fibration ) if itis local with respect to the set of maps S = { G ( n ) → F ( n ) : n ≥ } ∪ { F ( ) → E ( ) } . Definition 4.3.
We say a Reedy left (right) fibration Y → X is a left fibration ( right fibration ) if it is local withrespect to the set of maps S = { F ( ) → F ( n ) : n ≥ } .By Example 3.12 all maps in the set { G ( n ) → F ( n ) : n ≥ } ∪ { F ( ) → E ( ) } ∪ { F ( ) → F ( n ) : n ≥ } satisfy conditions (C) and (P) and so all results in Section 3 hold for their corresponding fibrations. In orderto summarize the results about the various localizations using the following table.Variance ( R ) Value ( V ) Fibration ( F ) Model Structure ( M ) Denoted ( D ) Reedy left
Seg
Segal coCartesian Segal coCartesian
SegcoCart
Reedy right
Seg
Segal Cartesian Segal Cartesian
SegCart
Reedy left
CSS coCartesian coCartesian coCart
Reedy right
CSS
Cartesian Cartesian
Cart
Reedy left
Kan left covariant cov
Reedy right
Kan right contravariant contra
We now have following results using the table above.
Theorem 4.4. (Theorem 3.5) There is a unique simplicial combinatorial left proper model structure on bisimplicialspaces over X, called the ( M ) -model structure and denoted by ( ss S / X ) ( D ) satisfying following conditions.(1) The fibrant objects are the ( F ) -fibrations over X.(2) Cofibrations are monomorphisms. (3) A map of bisimplicial spaces Y → Z over X is a weak equivalence if
Map / X ( Z , W ) → Map / X ( Y , W ) is a Kan equivalence for every ( F ) -fibration W → X.(4) A weak equivalence ( ( F ) -fibration) between fibrant objects is a level-wise equivalence (biReedy fibration). Proposition 4.5. (Proposition 3.6) The following adjunction ( ss S / X ) ( D ) ( ss S / X ) Diag − ( V ) id ⊥ id is a Quillen adjunction, which is a Quillen equivalence whenever X is homotopically constant. Here the left hand sidehas the ( M ) model structure and the left hand side has the induced diagonal ( V ) -model structure over the base X. Theorem 4.6. (Theorem 3.7) Let C be a small category. Then the adjunctions defined in Theorem 2.13 Fun ( C , s S ( V ) ) proj ( ss S / N C ) ( D ) Fun ( C , s S ( V ) ) projss R C ss H C ⊥ ss T C ss I C ⊥ are simplicial Quillen equivalences. Here the middle has the ( M ) -model structure and the two sides have the projectivemodel structure on the ( V ) -model structure. Theorem 4.7. (Corollary 3.16) Let p : L → X be a biReedy fibration. Then the following are equivalent:(1) p is an ( F ) -fibration.(2) p is an ( R ) -fibration and the simplicial space V al ( L ) is local with respect to ( V ) .(3) p is an ( R ) -fibration and the simplicial spaces V al k ( L ) are local with respect to ( V ) for all k ≥ .(4) For every map σ : F ( n ) × ∆ [ l ] → X, the pullback map σ ∗ p : σ ∗ L → F ( n ) × ∆ [ l ] is an ( F ) -fibration.(5) For every map σ : F ( n ) → X, the pullback map σ ∗ p : σ ∗ L → F ( n ) is an ( F ) -fibration.(6) p is an ( R ) -fibration and for every point { x } : F ( ) → X the fiber F ib x L is fibrant in the diagonal ( V ) -modelstructure.(7) p is an ( R ) -fibration and for every point { x } : F ( ) → X the fiber V al ( F ib x L ) is fibrant in the ( V ) -modelstructure. Theorem 4.8. (Theorem 3.17) Let L and M be two ( F ) -fibrations over X. Let g : L → M be a map over X. Then thefollowing are equivalent.(1) g : L → M is a biReedy equivalence.(2) V al ( g ) : V al ( L ) → V al ( M ) is a Reedy equivalence.(3) For every { x } : F ( ) → X, the map F ib x V al ( L ) → F ib x V al ( M ) is a Reedy equivalence of bisimplicialspaces.(4) For every { x } : F ( ) → X, the map F ib x ( L ) → F ib x ( M ) is a diagonal Reedy equivalence of bisimplicialspaces. For the next proposition we need following construction. Let p : W → X be an ( R ) -fibration. Then wecan construct following diagram(4.9) W • ˜ W • ˆ WX pi ≃ ˜ p j ≃ ˆ p .Here the first map is the level-wise functorial factorization of the simplicial object in ( s S / X ) ( V ) , the ( V ) -model structure. Moreover, let ˆ p : ˆ L → X be a biReedy fibrant replacement over X . Proposition 4.10. (Proposition 3.19) Let p : L → X, q : M → X be an ( R ) -fibrations and let f : L → M be a mapover X. Then the following are equivalent.
ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 25 (1) f is an ( M ) -equivalence.(2) The map ˆ f : ˆ L → ˆ M constructed in 4.9 is a biReedy equivalence.(3) The map V al ( ˆ f ) : V al ( ˆ L ) → V al ( ˆ M ) constructed in 4.9 is a Reedy equivalence.(4) The map V al ( f ) : V al ( L ) → V al ( M ) is a ( V ) -equivalence.(5) For every object { x } : F ( ) → X, the induced map on fibers V al ( F ib x L ) → V al ( F ib x M ) is a ( V ) -equivalence.(6) For every object { x } : F ( ) → X, the induced map on fibers F ib x L → F ib x Mis a diagonal ( V ) -equivalence. Theorem 4.11. (Theorem 3.20) A map g : Y → Z of bisimplicial spaces over X is an ( M ) -equivalence if and only iffor each map { x } : F ( ) → X, the induced mapY × X LF ib ( R x ) → Z × X LF ib ( R x ) is an equivalence in the diagonal ( V ) -model structure. Here R x is a choice of right fibrant replacement of the map { x } . Theorem 4.12. (Theorem 3.21) Let g : X → Y be a map of simplicial spaces. Then the adjunction ( ss S / X ) ( D ) ( ss S / Y ) ( D ) g ! ⊥ g ∗ is a Quillen adjunction, which is a Quillen equivalence whenever g is a CSS equivalence. Here both sides have the ( M ) -model structure. Theorem 4.13. (Theorem 3.22) Let p : V → X be a dual of an ( R ) -fibration over X. The induced adjunction ( ss S / X ) ( D ) ( ss S / X ) ( D ) p ! p ∗ ⊥ p ∗ p ∗ is a Quillen adjunction. Here both sides have the ( M ) -model structure. R EFERENCES[Ada95] J. F. Adams.
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ARTESIAN FIBRATIONS OF COMPLETE SEGAL SPACES 27 ´E COLE P OLYTECHNIQUE
F ´ ED ´ ERALE DE L AUSANNE , SV BMI UPHESS, S
TATION
8, CH-1015 L
AUSANNE , S
WITZERLAND
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