aa r X i v : . [ m a t h . C T ] F e b QUASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION
NIMA RASEKHA
BSTRACT . We prove that four different ways of defining Cartesian fibrations and the Cartesian model structureare all Quillen equivalent:(1) On marked simplicial sets (due to Lurie [Lur09a]),(2) On bisimplicial spaces (due to deBrito [BdB18]),(3) On bisimplicial sets,(4) On marked simplicial spaces.The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories andcomplete Segal spaces as defined by Joyal-Tierney and the straightening construction due to Lurie. C ONTENTS
0. Introduction 11. Review of Relevant Concepts 62. Marked Simplicial Spaces and Cartesian Fibrations 153. Equivalence of Cartesian Model Structures: Model Dependent and Independent 29References 31I
NTRODUCTION
For a given category C , we are often interested in studying pseudo-functors ofthe form P : C op → C atwhere C at is the (large) category of (small) categories. Such functors commonly arise in algebraic geometry(such as moduli problems [sga72]), topos theory (such as internal toposes [Joh02]), algebraic topology (suchas moduli stacks in chromatic homotopy [Con07]), ... .In certain mathematical situations, so-called derived or homotopical mathematics , the definition of a cate-gory is too strict and we need to work with a weaker notion of a category in which the usual axioms ofa category only hold up to suitably compatible equivalences or “homotopies” [Gro10]. The study of suchcategories is now usually called the theory of ( ∞ , 1 ) -categories [Ber10].Continuing our analogy with classical categories, the study of homotopical mathematics, such as derivedalgebraic geometry [Lur04], higher topos theory [Lur09a], ... , again requires us to study ( ∞ , 1 ) -functors P : C op → C at ( ∞ ,1 ) ,however, unlike the classical case, this is quite challenging and often not feasible. Indeed, because of thehigher coherence conditions, any such functor P requires an infinite tower of data, which usually preventsus from giving explicit descriptions. Date : February 2021.
Fortunately, there is an alternative approach towards such functors in the context of categories: fibrations .A functor p : D → C is called a Grothendieck fibration if every morphism in C , with specified lift of the targetin D , can be lifted to a p -Cartesian morphisms in D .Here, a morphism f : x → y in D is p -Cartesian if the induced commutative square D / y D / x C / p ( y ) C / p ( x ) f ∗ p pp ( f ) is a pseudo-pullback of categories.Using this definition of fibration, Grothendieck [gro03] proved following equivalence between pseudo-functors and Grothendieck fibrations, Z C : Fun ( C op , C at ) ≃ −→ G roth F ib C ,now commonly called the Grothendieck construction , thus giving us a way to translate between Grothendieckfibrations and pseudo-functors valued in categories.The fibrational approach is far more amenable to ( ∞ , 1 ) -categorical techniques and thus can be gener-alized in a straightforward manner. This was done by Lurie [Lur09a]. Using quasi-categories , which aresimplicial sets that model ( ∞ , 1 ) -categories [BV73, Joy08a, Joy08b], he defined Cartesian fibrations as innerfibrations of simplicial sets that have sufficient p -Cartesian lifts.Following a long tradition in homotopy theory, he then proceeds to define a model structure [Qui67] suchthat the fibrant objects are precisely the Cartesian fibrations over a fixed simplicial set S . However, he doesnot define the model structure on the category of simplicial sets over S , but rather on the category of markedsimplicial sets over S , which are simplicial sets with a chosen subset of the set of 1-simplices. Hence, despitethe fact that the definition of a Cartesian fibration requires no markings, the markings are a crucial aspectof its model structure. Using this model structure, aptly named the
Cartesian model structure , he then proves a Quillen equiva-lence [Lur09a, Theorem 3.2.0.1] ( s S et + / S ) Cart
Fun ( C [ S ] op , ( s S et + ) Cart ) proj St + S ⊥ Un + S which is simply a technical correspondence between Cartesian fibrations and functors valued in ( ∞ , 1 ) -categories, thus giving us a concrete way to transition between those, generalizing the Grothendieck con-struction from categories to ( ∞ , 1 ) -categories.In the subsequent years, several authors have studied Cartesian fibrations. For example, Riehl and Veritystudy Cartesian fibrations in the context of an ∞ - cosmos , which is a formal approach to ( ∞ , 1 ) -categories[RV17] motivated by formal category theory. Another example is the work by Ayala and Francis, whostudy the quasi-category of Cartesian fibrations from the perspective of exponentiable fibrations [AF20]. Itshould be noted, however, that neither approach results in a new model category for Cartesian fibrations.Rather they use the fact that the definition of ∞ -cosmoi or quasi-categories are less strict to study Cartesianfibrations directly, without having to take an external approach. It is in fact widely believed that it is not possible to define a model structure on simplicial sets over a given base, such that thefibrant objects are Cartesian fibrations and cofibrations are monomorphisms.
UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 3
In this paper we propose an alternative approachtowards Cartesian fibrations via complete Segal objects (rather than marked simplicial objects) which resultsin a new a model category of Cartesian fibrations. In order to understand this different approach it isnecessary to understand complete Segal spaces, right fibrations and how they can interact.A complete Segal space [Rez01] is a simplicial space W W W · · · that satisfies three sets of conditions (the Reedy, Segal and completeness conditions). These conditionsimply that complete Segal spaces are another model of ( ∞ , 1 ) -categories and in particular equivalent toquasi-categories [JT07] and can in fact be seen as the standard model [To¨e05].The benefit of complete Segal spaces is that the conditions can all be expressed via certain finite limitdiagrams [Rez10]. Hence, we can easily generalize a complete Segal space to a complete Segal object in anycategory (or model category or ( ∞ , 1 ) -category) with finite limits. Using our intuition from complete Segalspaces, we expect a complete Segal object in a category to play the role of an internal ( ∞ , 1 ) -category . So, inparticular, a complete Segal object in the category of space-valued presheaves is precisely a presheaf valuedin complete Segal spaces i.e. ( ∞ , 1 ) -categories.The discussion in the previous paragraph motivates us to study fibrations that correspond to presheavesvalued in spaces. As spaces are special kinds of ( ∞ , 1 ) -categories, we could simply think of them as specialkinds of Cartesian fibrations. However, it turns out there is a direct approach as well: right fibrations[Joy08a, Joy08b]. A right fibration is a map of simplicial sets that satisfies the right lifting condition withrespect to squares Λ [ n ] i R ∆ [ n ] S where n ≥
0, 0 < i ≤ n . Unlike Cartesian fibrations we can in fact endow the category of simplicial sets overa fixed simplicial set with a model structure, the contravariant model structure , such that the fibrant objectsare precisely the right fibrations. It was first proven by Lurie [Lur09a] (and later many other authors [Ste17,HM15, HM16]), that this model structure is Quillen equivalent to presheaves valued in spaces. Moreover,we can give an analogous definition of right fibrations and contravariant model structure in the context ofsimplicial spaces in a way that is Quillen equivalent to the contravariant model structure for simplicial sets[Ras17b, BdB18].Combining the two previous paragraphs, we should expect that a complete Segal object in right fibra-tions, which has the form R R R · · · W ,recovers functors valued in ( ∞ , 1 ) -categories. This intuition is in fact correct and the goal of this paper is tomake this statement precise and prove it. This idea is in fact not new. As an example, see the notion of a
Rezk object in [RV17].
NIMA RASEKH
Combining our previous observation, we are thus studying two ways of constructing Cartesian fibra-tions, the marked approach and the complete Segal approach, using two models of ( ∞ , 1 ) -categories, quasi-categories and complete Segal spaces, giving us a total of four characterizations of Cartesian fibrations,which we can depict as follows: Marked Approach Complete Segal Object ApproachQuasi-Categories ( s S et + / S ) Cart
Theorem 2.37 ( ss S et / S ) Cart
Theorem 1.35Complete Segal Spaces ( s S + / X ) Cart
Theorem 2.44 ( ss S / X ) Cart
Theorem 1.32The marked approach using quasi-categories is the original approach due to Lurie. The complete Segalapproach using complete Segal spaces is studied on its own in [Ras21], and directly results in the completeSegal approach using quasi-categories, which we cover in Subsection 1.6. This leaves us with the markedapproach using complete Segal spaces, which we study in Section 2. Finally, we show all four model struc-tures are indeed Quillen equivalent in Section 3.
Given that we already have a working definition ofCartesian fibrations using marked simplicial objects, why present an alternative approach using completeSegal objects?
Studying Cartesian Fibrations via Right Fibrations:
One set of applications follows from the key obser-vation that in the complete Segal approach Cartesian fibrations are a direct generalization of right fibrationsand thus, their properties can be directly deduced from right fibrations. • Generalized Cartesian Fibrations:
We can relax the complete Segal conditions used to define Carte-sian fibrations to define generalized Cartesian fibrations, that can be used to study presheaves val-ued in various other localizations of simplicial spaces, such as
Segal spaces [Ras21]. • Invariance of Cartesian Fibrations:
One important result about the Cartesian model structure isthe fact that it is invariant under categorical equivalences , meaning base change by a categorical equiv-alence gives us a Quillen equivalence. This is proven in [Lur09a, Proposition 3.3.1.1] using the factthat the Cartesian model structure is Quillen equivalent to a presheaf category.We can use a similar equivalence between the contravariant model structure and a presheaf cat-egory to similarly deduce that right fibrations are invariant under categorical equivalences. How-ever, in this case there is also an alternative proof, which uses the combinatorics of the contravariantmodel structure [HM15]. Using the complete Segal object approach to Cartesian fibrations we cangeneralize that proof immediately from right fibrations to Cartesian fibrations without having totranslate to presheaf categories. • Grothendieck Construction of Cartesian Fibrations:
Another example where we can exploit theconnection to right fibrations is the Grothendieck construction. In [Lur09a], Lurie first proves theequivalence between right fibrations and space-valued presheaves. He then wants to generalize thatto an equivalence between Cartesian fibrations and ( ∞ , 1 ) -category-valued presheaves, but cannotdirectly do so and thus needs several detailed and complicated proofs.On the other hand, in the complete Segal object approach to Cartesian fibrations every proof ofthe equivalence between right fibrations and space valued presheaves immediately generalizes to anew proof of the Grothendieck fibration for Cartesian fibrations. • Grothendieck Construction over -Categories: One particular instance of the previous point is thestudy of Cartesian fibrations over ordinary categories. In [HM15], the authors give a Grothendieckconstruction for right fibrations very much along the lines of the classical Grothendieck construc-tion for categories (also known as the category of elements ). They then prove that this gives us aQuillen equivalence. Again, using the complete Segal approach, we can realize that using the cate-gory of elements we can construct an equivalence between Cartesian fibrations over categories and
UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 5 presheaves. This construction is much simpler than the general unstraightening construction andeven simpler than the specific one given over categories in [Lur09a, 3.2.5].
Studying Fibrations of ( ∞ , n ) -Categories: The way we generalized 1-categories to ( ∞ , 1 ) -categories, wecan generalize strict n -categories to ( ∞ , n ) -categories [Ber20]. Moreover, similar to the ( ∞ , 1 ) -categoricalcase, the study of functors P : C op → C at ( ∞ , n ) ,where C is an ( ∞ , n ) -category, is quite a challenge, as we encounter the same problems with homotopycoherence. Thus we again would like to develop an appropriate theory of fibrations.However, the ( ∞ , n ) -categorical case faces far more challenges than the ( ∞ , 1 ) -categorical case. First ofall, similar to the ( ∞ , 1 ) -categories, there are several models of ( ∞ , n ) -categories: θ n -spaces [Rez10], θ n -sets [Ara14], n-fold complete Segal spaces [Bar05], complicial sets [Ver08], comical sets [CKM20], ... . However, it isnot known whether these models are in fact equivalent (except for θ n -sets, θ n -spaces and n -fold completeSegal spaces [BR13, BR20, Ara14]). Second, we don’t have a general definition of fibrations for any of thesemodels of ( ∞ , n ) -categories. Hence, unlike for ( ∞ , 1 ) -categories, we cannot just focus on one model, prove all the relevant results, andthen translate the results, via a web of Quillen equivalences. If we need a theory of fibrations for a certainmodel, then we really need to study it on its noted.Finally, we also cannot just ignore certain models, with the hope that future results will facilitate trans-lating results from one model to another. First of all it is not certain that such equivalences of modelswill be developed anytime soon (even the study of strict n -categories is quite challenging) and second ofall most models already have concrete that applications in other branches of mathematics. For example n -fold complete Segal spaces are currently the primary method for the study of topological field theories[Lur09c, CS19]. Or 2-complicial sets (or alternatively 2-comical sets) have found applications in the studyof derived algebraic geometry [GR17a, GR17b].As θ n -spaces and n -fold complete Segal spaces are directly generalizations of complete Segal spaces, wewould expect that a complete Segal approach towards Cartesian fibrations can be a powerful tool in thestudy of their fibrations. There are several interesting question about the connection between thevarious Cartesian model structures that remains unexplored:(1) The equivalence between the Cartesian model structures relies on proving that all four are equiv-alent to various functor categories. It is not clear whether we can construct a direct equivalencebetween marked simplicial objects and bisimplicial objects that induces an equivalence between theassociated Cartesian model structures.(2) Along the same lines, one of the benefits of using the marked simplicial approach is that it is rea-sonably easy to see that Cartesian fibrations are themselves categorical fibrations. We would like togeneralize this result and, for example, prove that generalized Cartesian fibrations that characterizefunctors valued in Segal spaces are always Segal fibrations [Ras21]. However, the current Quillenequivalences do not allow us to draw such a conclusion.Answering this question might rely on first having a better way to translate between markedsimplicial objects and bisimplicial objects.(3) Assuming we can construct a direct equivalence between marked simplicial sets and bisimplicialsets, a final question would be whether we can effectively use that to deduce all the results aboutCartesian fibrations as originally proven in [Lur09a, Chapter 3], hence significantly simplifying theresults. There are some results about fibrations of 2-complicial sets [Lur09b], there called scaled simplicial sets . NIMA RASEKH
This paper is the second part of a three-paper series which introduces thebisimplicial approach to Cartesian fibrations:(1)
Cartesian Fibrations of Complete Segal Spaces [Ras21](2)
Quasi-Categories vs. Segal Spaces: Cartesian Edition (3)
Cartesian Fibrations and Representability [Ras17a]In particular, the first paper includes a detailed analysis of the Cartesian model structure on bisimplicialspaces, which we only review here (Subsection 1.6). The third paper gives an application of the bisimplicialapproach to the study of representable Cartesian fibrations.
Given that the goal is to prove existence of several Quillen equivalences, we use manydifferent model structures (some times on the same category). Thus, in order to avoid any confusion, wewill denote every category with its associated model structure. In order to help the reader, here is a list of allrelevant model structures (except the four Cartesian model structures already mentioned), the underlyingcategory, along with the abbreviation and a reference to their definition:
Model Structure Category(ies) Abbreviation Reference
Joyal Model Structure s S et Joy
Theorem 1.12Complete Segal Space Model Structure s S CSS
Theorem 1.16Complete Segal Object Model Structure s M CSO
Theorem 1.19Contravariant Model Structure s S / X contra Theorem 1.26Contravariant Model Structure s S et / S contra Theorem 1.28localized unmarked Reedy Model Structure s S + un + Ree L Proposition 2.19unmarked CSS Model Structure s S + un + CSS
Corollary 2.22marked CSS Model Structure s S + CSS + Theorem 2.26
I would like to thank Charles Rezk for suggesting the study of Cartesian fibra-tions of simplicial spaces, which has been the main motivation for this work. I would also like to thankJoyal and Tierney for their beautiful work in [JT07], which is the theoretical backbone of this paper and hasbeen the motivation for the title. R
EVIEW OF R ELEVANT C ONCEPTS
In this section we review some important concepts and establish necessary notation.
Let ∆ be the simplex category with object posets [ n ] = {
0, 1, ..., n } and morphisms maps of posets. We will use a variety of simplicial objects, i.e. functors X : ∆ op → C andthus need to distinguish them carefully, based on the value of the simplicial objects. Simplicial Sets/Spaces:
We will use two terminologies for functors X : ∆ op → S et: • It is a space if it is an object in the category of simplicial sets with the Kan model structure. In thatcase the category of spaces is denote S . • It is a simplicial set if it is an object in the category of simplicial sets with any other model structure(such as Joyal, contravariant, ...). In that case the category of simplicial sets is denoted by s S et.We will use following notation regarding simplicial sets/spaces:(1) ∆ [ n ] denotes the simplicial set representing [ n ] i.e. ∆ [ n ] k = Hom ∆ ([ k ] , [ n ]) .(2) ∂ ∆ [ n ] denotes the boundary of ∆ [ n ] i.e. the largest sub-simplicial set which does not include id [ n ] : [ n ] → [ n ] . Similarly Λ [ n ] l denotes the largest simplicial set in ∆ [ n ] which does not include l th face. UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 7 (3) 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. Bisimplicial Sets/Simplicial Spaces:
Similar to above we will use two terminologies for functors X : ∆ op × ∆ op → S et: • It is a simplicial space if it is an object in the category of bisimplicial sets with the Reedy modelstructure or any localization thereof. In that case the category of simplicial spaces is denote s S . • It is a bisimplicial set if it is an object in the category of bisimplicial sets with any other model structure(such as Reedy contravariant, ...). In that case the category of bisimplicial sets is denoted by ss S et.We will use following notation convention for simplicial spaces.(1) Denote F ( n ) to be the simplicial space defined as F ( n ) kl = Hom ∆ ([ k ] , [ n ]) .Similarly, we define the simplicial space ∆ [ l ] as ∆ [ n ] kl = Hom ∆ ([ l , [ n ]) Note, F ( n ) × ∆ [ n ] generate the category of simplicial spaces.(2) 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).(3) ∂ F ( n ) denotes the boundary of F ( n ) . Similarly L ( n ) l denotes the largest simplicial space in F ( n ) which lacks the l th face.(4) The category s S is enriched over spacesMap s S ( X , Y ) n = Hom s S ( X × ∆ [ n ] , Y ) .(5) The category s S is also enriched over itself: ( Y X ) kn = Hom s S ( X × F ( n ) × ∆ [ l ] , Y ) .(6) By the Yoneda lemma, for a simplicial space X we have a bijection of spaces X n ∼ = Map s S ( F ( n ) , X ) . Bisimplicial Spaces:
We also use functors X : ∆ op × ∆ op × ∆ op → S et, which we call bisimplicial space and denote by ss S . In this section we review some of the underlying model structuresthat we can define on the categories defined in Subsection 1.1.
Kan Model Structure:
The
Kan model structure is one of the first model structures defined and can alreadybe found in [Qui67]. For another detailed account of the Kan model structure see [GJ99].
Theorem 1.1.
There is a unique simplicial, combinatorial, proper model structure on S , called the Kan model struc-ture and denoted S Kan characterized as follows:(1) A map is a cofibration if it is a monomorphism.(2) A map is a fibration if satisfies the lifting right lifting property with respect to horns Λ [ n ] i → ∆ [ n ] .(3) A map is a weak equivalence if it is homotopy equivalence. (its geometric realization is an equivalence oftopological spaces). Reedy Model Structure for Simplicial Objects:
Let C M be a model category with model structure M .Then we can define a new model structure on the category of simplicial objects Fun ( ∆ op , C ) = s C . Thismodel structure exists for a wide range of model categories, however, here we will focus on the case of NIMA RASEKH interest to us. It was originally constructed by Reedy [Ree74], however we will use [Hir03] as our primaryreference.
Theorem 1.2. [Hir03, Theorem 15.3.4, Proposition 15.6.3]
Let C M be a combinatorial, simplicial, left propermodel structure such that the cofibrations are monomorphisms. Then there exists a simplicial, combinatorial, leftproper model structure on simplicial objects in C , which we call the Reedy model structure and denote s C Ree M , whichhas following characteristics:(1) A map p : Y → X is a Reedy cofibration if it is a monomorphism.(2) A map p : Y → X is a Reedy weak equivalence if it is level-wise a weak equivalence in C M .(3) A map p : Y → X is a Reedy fibration if for every n ≥ the induced mapY n → M n Y × M n X X n is a fibration in C M . Here M n is the n-th matching object [Hir03, Definition 15.2.5] .Remark . If C M = S Kan , then the n -th matching object of a simplicial space X is given by the spaceM n X = Map s S ( ∂ F ( n ) , X ) [Hir03, Proposition 15.6.15].Notice the construction is invariant under Quillen equivalences. Lemma 1.4. [Hir03, Proposition 15.4.1]
Let C M D N F ⊥ G be a Quillen equivalence between model categories. Then the induced adjunction ( s C ) Ree M ( s D ) Ree N sF ⊥ sG is a Quillen equivalence as well. Projective Model Structure for Functor Categories
The existence of the Reedy model structure relied onthe fact that ∆ is a Reedy category and so the Reedy model structure cannot be applied to any functor cat-egory Fun ( C , D ) . Fortunately, under mild assumptions on a model category, we can construct the projectivemodel structure . Theorem 1.5. [Hir03, Theorem 11.6.1, Theorem 11.7.3]
Let C be a small category and D M a combinatorial,simplicial, left proper model category. Then there exists a unique left proper, simplicial, combinatorial model structureon the functor category, denoted Fun ( C , D M ) proj and called the projective model structure, such that(1) A map α : F → G is a projective fibration if for all objects c in C the map α c : F ( c ) → G ( c ) is a fibration in D M .(2) A map α : F → G is a projective weak equivalence if for all objects c the map α c : F ( c ) → G ( c ) is a weakequivalence in D M . Similar to the Reedy model structure the projective model structure is also invariant.
Lemma 1.6. [Hir03, Theorem 11.6.5]
Let D M E N F ⊥ G be a Quillen equivalence between combinatorial simplicial left proper model categories and let C be a small category.Then the induced adjunction UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 9
Fun ( C , D M ) proj Fun ( C , E N ) projF ∗ ⊥ G ∗ is a Quillen equivalence as well. In this subsection we review several important (and technical)results that help us construct new model structures.One very common technique is known as Bousfield localization.
Theorem 1.7.
Let C M be a category with a left proper, combinatorial, simplicial model structure M . Moreover, let L be a set of cofibrations in C M . There exists a unique cofibrantly generated, left proper, simplicial model categorystructure on C , called the L -localized model structure and denoted C M L with the following properties:(1) A morphism Y → Z is a cofibration in C M L if it is a cofibration in C M .(2) An object W is fibrant (also called L -local) if it is fibrant in C M and for every morphism f : A → B in L ,the induced map Map C ( B , W ) → Map C ( A , W ) is a Kan equivalence.(3) A map Y → Z is a weak equivalence in M L if for every L -local object W the induced map Map C ( Z , W ) → Map C ( Y , W ) is a Kan equivalence. The existence of the L -localized model structure is proven using the theory of left Bousfield localizations.For a careful and detailed proof of the existence of left Bousfield localizations see [Hir03, Theorem 4.1.1]For a nice summary of this proof that goes over the main steps see [Rez01, Proposition 9.1].Using a variation of Bousfield localization and ideas from [Qui67] get right induced model structures. Theorem 1.8.
Let D M C F ⊥ G be an adjunction, such that the following conditions hold:(1) C is a simplicial, locally presentable category.(2) D M is a simplicial, left proper, combinatorial model category(3) G preserves filtered colimits(4) C has a functorial fibrant replacement.Then there exists a unique simplicial, combinatorial, left proper model structure on C N , called the right induced modelstructure, and characterized by the fact that a map f : c → d in C N is a fibration (weak equivalence) if and only if G fis a fibration (weak equivalence) in D M . The proof is a combination of results [GS07, Theorem 3.6, Theorem 3.8, Corollary 4.14].There is also another version of a Bousfield localization we need.
Theorem 1.9. [Lur09a, A.3.7.10]
Let C M and D N be left proper combinatorial simplicial model categories such thatcofibrations on in C M and D N are monomorphisms and suppose we are given a simplicial Quillen adjunction C M D N F ⊥ G Then(1) There exists a new left proper combinatorial simplicial model structure C M ′ on the category C with thefollowing properties: • A morphism f in C M ′ is a cofibration if and only if it is a cofibration in C M . • A morphism f in C M ′ is a weak equivalence if and only if F f is a weak equivalence in D N . • A morphism in f in C M ′ is a fibration if and only if it has the right lifting property with respect to trivialcofibrations.(2) The functors F and G determine a new simplicial Quillen adjunction C M ′ D N F ⊥ G (3) If the the derived right adjoint is fully faithful then the Quillen adjunction is a Quillen equivalence. Finally, we want to observe how a Quillen equivalence of model categories can be transferred to over-categories.
Proposition 1.10.
Let C M and D N be two model categories in which all objects are cofibrant and let C M D N F ⊥ G be a Quillen adjunction. Fix an object C ∈ C . Then the adjunction ( C / C ) M ( D / FC ) N F ⊥ u ∗ G is a Quillen adjunction. Here F ( C ′ → C ) = FC ′ → FC and u ∗ G ( D → FC ) = ( u C ) ∗ ( GD → GFC ) whereu C : C → GFC is the unit map of the adjunction. Moreover, if ( F , G ) is simplicial (or a Quillen equivalence) then ( F , u ∗ G ) is also simplicial (or a Quillen equivalence). ( ∞ , 1 ) -Categories and their Equivalence. There are now many different ap-proaches to the theory of ( ∞ , 1 ) -categories. Here we focus on two very popular models. Quasi-categoriesand complete Segal spaces.Quasi-categories were first introduced by Boardman and Vogt in their study of homotopy coherent alge-braic structures [BV73]. It was Joyal who realized that quasi-categories can be used to do concrete categorytheory [Joy08a, Joy08b] and then later Lurie, who developed all of category theory in the context of quasi-categories [Lur09a, Lur17]. Here we only review the definition and the existence of a model structure. Definition 1.11. [Lur09a, Definition 1.1.2.4] A quasi-category is a simplicial set S that has right lifting prop-erty with respect to diagrams Λ [ n ] i S ∆ [ n ] where 0 < i < n .Quasi-categories come with a model structure. Theorem 1.12. [Lur09a, Theorem 2.2.5.1]
There is a unique, combinatorial, left proper model structure on s S et,called the Joyal model structure and denoted s S et Joy , with the following specifications:
UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 11 • A map S → is a cofibration if it is a monomorphism. • An object S is fibrant if it is a quasi-category.
Another popular model of ( ∞ , 1 ) -categories are complete Segal spaces, which were defined and studiedby Charles Rezk [Rez01]. Definition 1.13.
Let G ( n ) be the sub-simplicial space of F ( n ) consisting of maps f : [ k ] → [ n ] such that f ( k ) − f ( ) ≤
1. Define the sets S eg and C omp as follows: S eg = { G ( n ) ֒ → F ( n ) : n ≥ }C omp = { F ( ) → E ( ) } Notation . We denote the subobject of ∆ [ n ] that is defined similarly to G ( n ) by Sp [ n ] . Definition 1.15. [Rez01, 6] A Reedy fibrant simplicial space W is called a complete Segal space if for everymap f : A → B in S eg ∪ C omp, the induced mapMap s S ( B , W ) → Map s S ( A , W ) is a Kan equivalence.The definition already suggests that complete Segal spaces are fibrant objects in a model structure. Theorem 1.16. [Rez01, Theorem 7.2]
There is a unique simplicial combinatorial, left proper model structure onthe category of simplicial spaces, called the complete Segal space model structure and denoted by s S CSS , given by thefollowing specifications:(1) A map f : Y → Z is a cofibration if it is a monomorphism.(2) An object W is fibrant if it is a complete Segal space.(3) A map f : Y → Z is a weak equivalence if for every complete Segal space W the induced map
Map s S ( Z , W ) → Map s S ( Y , W ) is a Kan equivalence. Before we proceed to the comparison between complete Segal spaces and quasi-categories, we want togeneralize complete Segal spaces to complete Segal objects.Let C M be a model category. The fact that C has small coproducts implies that we have a functor C op : S et → C ,uniquely characterized by the fact that it takes the one element set to the final object in C . This can beextended to a functor of simplicial objects: s C op : s S et → s C Now we can use this notation for following definition.
Definition 1.17.
For a given category C define the sets S eg M and C omp M as S eg M = { s C op ( Sp [ n ] ֒ → ∆ [ n ]) : n ≥ }C omp M = { s C op ( ∆ [ ] ֒ → J [ ]) } Definition 1.18.
Let C M be a combinatorial, simplicial, left proper model category such that the cofibrationsare the monomorphisms. A functor W : ∆ op → C is called a complete Segal object if it satisfies followingconditions:(1) It is fibrant in s C Ree M .(2) For every morphism A → B in S eg M ∪ C omp M and every object K in C the induced mapMap s M ( B × K , W ) → Map s M ( A × K , W ) is a Kan equivalence. Theorem 1.19.
Let C M be a combinatorial, simplicial, left proper model category such that the cofibrations aremonomorphisms. Then there exists a unique combinatorial, simplicial, left proper model category structure on thecategory s C , denoted s C CSO M and called the complete Segal object model structure, such that it satisfies followingconditions:(1) A map A → B is a cofibration if it is a monomorphism of simplicial objects.(2) An object W is fibrant if it is a complete Segal object.(3) A map of simplicial objects Y → Z is a weak equivalence if the induced map
Map s C ( Z , W ) → Map s C ( Y , W ) is a Kan equivalence for every complete Segal object W.We will often simplify the notation to s C CSO , if the model structure M is clear from the context.Proof. By Theorem 1.2, s C Ree M is also combinatorial, simplicial and left proper and where the cofibrationsare monomorphisms. In particular, s C is locally presentable and so we can choose a get of generators I Hence, by Theorem 1.7, we can construct the localized model structure s C ( Ree M ) L , where L = { A × K → B × K | A → B ∈ S eg M ∪ C omp M , K ∈}S eg M ∪ C omp M , K ∈ I} which immediately satisfies the desired results. (cid:3) Remark . See [RV17, Proposition 2.2.9] for more details about the complete Segal object model structure(there called
Rezk object ) and other properties it inherits from C M .Finally, the complete Segal objects are also homotopy invariant. Theorem 1.21.
Let C M D N F ⊥ G be a Quillen equivalence of combinatorial, simplicial, left proper model structures. Then the induced adjunction ( s C ) CSO M ( s D ) CSO N sF ⊥ sG is also a Quillen equivalence.Proof. This follows directly from combining Lemma 1.4 and Theorem 1.9. (cid:3)
Until now we have claimed that quasi-categories and complete Segal spaces are two models of ( ∞ , 1 ) -categories. This requires knowing that they are in fact equivalent. This was proven by Joyal and Tierney[JT07], who constructed two Quillen equivalences between complete Segal spaces and quasi-categories thatwill play an important role later on and hence we will review here.Let p : ∆ × ∆ → ∆ be the projection functor that takes ([ n ] , [ m ]) to [ n ] .Similarly, let i : ∆ → ∆ × ∆ be the inclusion functor that takes [ n ] to ([ n ] , [ ]) . Theorem 1.22. [JT07, Theorem 4.11]
The induced adjunction
UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 13 s S et Joy s S CSSp ∗ ⊥ i ∗ is a Quillen equivalence between the Joyal model structure and the complete Segal space model structure. We move on to the second Quillen equivalence. Let t : ∆ × ∆ → s S etbe the functor defined as t ([ n ] , [ m ]) = ∆ [ n ] × J [ m ] . Theorem 1.23. [JT07, Theorem 4.12]
Let s S CSS s S et Joyt ! ⊥ t ! be the adjunction induced by the map t (meaning t ! is the left Kan extension). Then this adjunction is a Quillenequivalence between the complete Segal space model structure and the Joyal model structure.Remark . The adjunction ( t ! , t ! ) is not simplicial adjunction (in fact the Joyal model structure is not evensimplicial). However, the adjunction is enriched over the Joyal model structure. So, in particular, for twoquasi-categories S , T we have an equivalence of complete Segal spaces t ! ( T S ) → t ! T t ! S For a detailed discussion of this enrichment see [RV17, Example 2.2.6]The goal is to show that with some minor adjustments we can generalize these two Quillen equivalencesto Quillen equivalences between various notions of Cartesian fibrations.
In this subsection we define the contravariant model structure forsimplicial spaces and simplicial sets and observe that they are in fact Quillen equivalent.Let us start with the case of simplicial spaces.
Definition 1.25. [Ras17b, Definition 3.2, Remark 4.24] Let X be a simplicial space. A map p : R → X iscalled right fibration if it is a Reedy fibration and the following is a homotopy pullback square: R n X n R X < n > ∗ p n p p < n > ∗ .where < n > : F ( ) → F ( n ) is the map that takes the point to n ∈ F ( n ) .Right fibrations come with a model structure. Theorem 1.26. [Ras17b, Theorem 3.12, Remark 4.24]
Let X be simplicial space. There is a unique simplicial,combinatorial, left proper model structure on the category s S / X , called the contravariant model structure and denotedby ( s S / X ) contra , which satisfies the following conditions:(1) An object R → X is fibrant if it is a right fibration.(2) A map Y → Z over X is a cofibration if it is a monomorphism. (3) A map f : Y → Z over X is a weak equivalence if
Map / X ( Z , R ) → Map / X ( Y , R ) is an equivalence for every right fibration R → X. Now we can move on to the case of simplicial sets.
Definition 1.27. [Lur09a, Definition 2.0.0.3] A map of simplicial sets p : R → S is a right fibration if it hasthe right lifting property with respect to squares Λ [ n ] i R ∆ [ n ] S p where n ≥ < i ≤ n .Similar to above, we can define a model structure with fibrant objects right fibrations. Theorem 1.28. [Lur09a, Proposition 2.1.4.7, Proposition 2.1.4.8, Proposition 2.1.4.9, Remark 2.1.4.12]
Let Sbe a simplicial set. There is a unique simplicial, combinatorial, left proper model structure on the category of simplicialsets over S, called the contravariant model structure and denoted by ( s S et / S ) contra , such that(1) An object R → S is fibrant if it is a right fibration.(2) A map T → U over S is a cofibration if it is a monomorphism.(3) A map T → U over S is a weak equivalence if the induced map
Map / S ( U , R ) → Map / S ( T , R ) is a Kan equivalence for every right fibration R → S. Notice, we called both model structures (Theorem 1.28, Theorem 1.26) contravariant, as they are in factQuillen equivalent.
Theorem 1.29. [Ras17b, Theorem B.12, Theorem B.13]
Let S be a simplicial set and X a simplicial space. Thereare Quillen equivalences of contravariant model structures ( s S / X ) contra ( s S et / t ! X ) contrat ! ⊥ u ∗ t ! ( s S et / S ) contra ( s S / p ∗ S ) contrap ∗ ⊥ i ∗ In this subsection we review the complete Segal ap-proach towards Cartesian fibrations and then prove the analogous results for Cartesian fibrations of bisim-plicial sets. The case for bisimplicial spaces has been studied in great detail in [Ras21].Let X be a simplicial space. We denote by X the bisimplicial space defined as X kn = X n . Definition 1.30. [Ras21, Section 4] Let X be a simplicial space. A map of bisimplicial spaces Y → X is aCartesian fibration, if it satisfies following three conditions:(1) Y → X is a Reedy fibration in ss S .(2) For each k , the map of simplicial spaces Y k → X is a right fibration.(3) The simplicial space Y • is a complete Segal space. Remark . As X is a simplicial space, we have an isomorphism of categories ss S / X ∼ = Fun ( ∆ op , s S / X ) Using this isomorphism we can fact replace condition ( ) with condition UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 15 (3’) The corresponding simplicial functor Y : ∆ op → s S / X is a complete Segal object in the contravariant model structure on s S / X . Theorem 1.32. [Ras21, Section 4]
Let X be a simplicial space. There is a unique simplicial combinatorial left propermodel structure on ss S / X , called the Cartesian model structure and denoted ( ss S / X ) Cart , such that • An object C → X is fibrant if it is a Cartesian fibration (Definition 1.30). • A map Y → Z over X is a cofibration if it is a monomorphism. • A map Y → Z over X is a weak equivalence if for every Cartesian fibration C → X, the induced map
Map / X ( Z , C ) → Map / X ( Y , C ) is a Kan equivalence.Remark . It follows from Remark 1.31 that this model structure is in fact the complete Segal object modelstructure on the contravariant model structure on simplicial spaces over X , as constructed in Theorem 1.19.For more details about the Cartesian model structure (in particular various ways of characterizing itsfibrant objects and weak equivalences) see [Ras21].We now move on to the analogous result for Cartesian fibrations of bisimplicial sets. For a given simpli-cial set S , we denote by S the bisimplicial set defined as S kn = S n . Definition 1.34.
An object T → S in ss S et / S is a Cartesian fibration if the corresponding functor T : ∆ op → ( s S et / S ) contra is a complete Segal object (Definition 1.18) in the contravariant model structure on ( s S et / S ) contra .We want an analogous result about Cartesian fibrations of bisimplicial sets. Here we can simply combineTheorem 1.21, Theorem 1.26, and Theorem 1.19 to get following extensive result. Theorem 1.35.
Let S be a simplicial sets. There exists a unique, simplicial, combinatorial, left proper model structureon the category of bisimplicial sets over S, ss S et / S , called the Cartesian model structure and denoted ( ss S et / S ) Cart with following properties:(1) An object C → S is fibrant if it is a Cartesian fibration (Definition 1.34).(2) A map of bisimplicial sets T → U over S is a cofibration if it is a monomorphism.(3) A map T → U over S is a weak equivalence if for every Cartesian fibration C → S, the induced map
Map / S ( U , C ) → Map / S ( T , C ) is a Kan equivalence.Moreover, we have Quillen equivalences: ( ss S / X ) Cart ( ss S et / st ! X ) Cartst ! ⊥ u ∗ st ! ( ss S et / S ) Cart ( ss S / sp ∗ S ) Cartsp ∗ ⊥ u ∗ si ∗ . M ARKED S IMPLICIAL S PACES AND C ARTESIAN F IBRATIONS
In Subsection 1.6 we defined two Cartesian model structures: for bisimplicial spaces and for bisimplicialsets. As mentioned before, the Cartesian model structure for marked simplicial sets had already beendefined [Lur09a]. This leaves us with one last Cartesian model structure: marked simplicial spaces , which isthe goal of this section.
In this subsection we want to study marked simplicial spaces. We firstreview marked simplicial sets as studied in [Lur09a].
Definition 2.1. A marked simplicial set is a pair ( S , A ) where S is a simplicial set and A ⊂ S such that A includes all degenerate edges. A morphism of marked simplicial sets f : ( S , A ) → ( T , B ) is a map ofsimplicial sets f : S → T such that f ( A ) ⊂ B . We denote the category of marked simplicial sets by s S et + .We want to study the category of marked simplicial sets s S et + . As we the additional data of markings,the objects are not just simplicial objects. We thus need an alternative approach. Definition 2.2.
Let ∆ + , the marked simplex category , be the category defined as the pushout { [ ] → [ ] } { [ ] → [ + ] → [ ] } ∆ ∆ + where the top and left hand maps are the evident inclusion maps (note there is a unique map [ ] → [ ] in ∆ ).Using our intuition about ∆ , we can depict this category as follows: [ ] [ ] [ ] · · · [ + ] where the evident triangle is commutative.We would like to prove that a marked simplicial set is just a presheaf on ∆ + , where the image of [ + ] is the set of markings. However, for that we need the additional assumption that the image of the map [ + ] → [ ] is an injection of sets. Hence, marked simplicial sets are certain separated presheaves on ∆ + . Lemma 2.3.
Let J be the Grothendieck topology on ∆ + where the only non-trivial cover is [ ] → [ + ] . Then thecategory of marked simplicial sets is isomorphic to the category of separated set valued presheaves on ∆ + with thetopology J .Proof. By definition F : ∆ + → S et is separated if it takes covering maps to monomorphisms. Hence, F isseparated with respect to the topology J if and only if F ([ + ]) → F ([ ]) is a monomorphism which meansit is a marked simplicial set, where the markings are F ([ + ]) .It is now immediate to confirm that a natural transformation of functors corresponds to maps of markedsimplicial sets. (cid:3) Remark . This lemma is also proven (independently) in [nla20].
Remark . This lemma proves that the category of marked simplicial sets is equivalent to a category ofseparated presheaves, meaning it is a quasi-Grothendieck topos [Joh02, C2.2.13], which has far-reachingimplications:(1) s S et + has small limits and colimits(2) s S et + is a presentable category.(3) s S et + is locally Cartesian closed, which for two objects ( X , A ) , ( Y , B ) we denote by ( Y , B ) ( X , A ) . UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 17
Notation . We will make ample use of the fact that marked simplicial objects are subcategories of presheafcategories. Hence we generalize our notational conventions from Subsection 1.1 and denote the generatorsof Fun (( ∆ + ) op , S et ) by ∆ + [ n ] . Notice, here n is an object in ∆ + and so we can also have n = + .Finally, we need to realize how marked simplicial sets are related to simplicial sets [Lur09a, Definition3.1.0.1]. Remark . There is a functor inc : ∆ → ∆ + , which is the identity on objects and gives us a chain ofadjunctions Fun (( ∆ + ) op , S et ) Fun ( ∆ op , S et ) inc ∗ inc ! ⊥ inc ∗ ⊥ ,which restricts to adjunctions s S et + s S et Unm ( − ) ♭ ⊥ ( − ) ⊥ .Here Unm simply forgets the markings and, for a given simplicial set S , we have S ♭ = ( S , S ) and S =( S , S ) . Remark . We can use the standard simplices ∆ [ n ] and the functors ( − ) ♭ and ( − ) ♯ to get following valuableisomorphisms: Hom s S et + ( ∆ [ n ] ♭ , ( S , A )) ∼ = S n ,Hom s S et + ( ∆ [ ] ♯ , ( S , A )) ∼ = A ,for all marked simplicial sets ( S , A ) and n ≥ Definition 2.9. A marked simplicial space is a pair ( X , A ) , where X is a simplicial space and A ֒ → X is aninclusion of spaces such that the degeneracy map s : X → X takes value in A . We say X is the underlyingsimplicial space and A is the space of markings . Definition 2.10.
A map of marked simplicial spaces f : ( X , A ) → ( Y , B ) is a map of simplicial spaces f : X → Y such that f restricts to a map ( f ) | A : A → B . Marked simplicial spaces with the morphismsdescribed before form a category which we denote by s S + .We now want to proceed to study the category of marked simplicial spaces. Comparing Definition 2.1and Definition 2.9 it immediately follows that a marked simplicial space is just a simplicial object in thecategory of marked simplicial sets. Hence, we get the following lemma that allows us to straightforwardlygeneralize results from marked simplicial sets to marked simplicial spaces. Lemma 2.11.
There is an equivalence of categories
Fun ( ∆ , s S et + ) ∼ = s S + .Remark . The equivalence immediately has following implications:(1) s S + has small limits and colimits and presentable.(2) s S + is locally Cartesian closed, which for two objects ( X , A ) , ( Y , B ) we denote by ( Y , B ) ( X , A ) .(3) Similarly, we denote the generators of Fun ( ∆ + , S ) by F + ( n ) × ∆ [ l ] . Notice, here n is again an objectin ∆ + , whereas l is an object in ∆ . (4) There are adjunctions s S + s S Unm ( − ) ♭ ⊥ ( − ) ⊥ .Here Unm simply forgets the markings and, for a given simplicial space X , we have X ♭ = ( X , X ) and X = ( X , X ) .(5) Using Unm we can define a simplicial enrichmentMap s S + (( X , A ) , ( Y , B )) = Unm (( Y , B ) ( X , A ) ) .(6) Finally, we have Map s S + ( F ( n ) ♭ , ( X , A )) ∼ = X n ,Map s S + ( F ( ) ♯ , ( X , A )) ∼ = A ,for all marked simplicial spaces ( X , A ) and n ≥ In this subsection we want togeneralize the Joyal and complete Segal space model structures to marked simplicial sets and spaces. Thefirst step is to adjust the Joyal-Tierney adjunctions (Theorem 1.22,Theorem 1.23) to marked objects.Let i + : ∆ + → ∆ + × ∆ be the map that takes an object [ n ] to ([ n ] , [ ]) . Similarly, let p + : ∆ + × ∆ → ∆ + be the functor that takes a pair ([ n ] , [ m ]) to [ n ] . Definition 2.13.
Let Fun ( ∆ + , S et ) Fun ( ∆ + × ∆ , S et ) ( p + ) ∗ ⊥ ( i + ) ∗ be the adjunction induced by the maps i + , p + .We claim this adjunction restricts to marked objects. Lemma 2.14.
The functors ( p + ) ∗ , ( i + ) ∗ take marked objects to marked object. Hence, they restrict to an adjunctions S et + s S +( p + ) ∗ ⊥ ( i + ) ∗ .Proof. Let G : ( ∆ + ) op → S et be a presheaf that corresponds to a marked simplicial set, meaning the map G ( + ) → G ( ) is a monomorphism. We need to show that ( p + ) ∗ ( G )( + , l ) → ( p + ) ∗ ( G )( l ) is a monomor-phism as well. However, by direct computation ( p + ) ∗ ( G )( + , l ) → ( p + ) ∗ ( G )( l ) = G ( + ) → G ( ) andso the result follows from our assumption.On the other hand, assume G : ( ∆ + ) op × ∆ op → S et be a presheaf that corresponds to marked simplicialspace. We want to prove ( i + ) ∗ ( G ) : ( ∆ + ) op → S et corresponds to a marked simplicial set, meaning wehave to show ( i + ) ∗ ( G )( + ) → ( i + ) ∗ ( G )( ) is a monomorphism. By direct computation, this map is givenby G ( + , 0 ) → G (
1, 0 ) , which is a monomorphism by assumption. (cid:3) UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 19
We now move on to adjust the second adjunction, ( t ! , t ! ) , to the marked setting.Let τ : ∆ + → s S et + be the marked simplicial diagram defined as τ ([ n ]) = ( ∆ [ n ] ♭ if n = + ∆ [ ] ♯ if n = + ,where the cosimplicial maps between ∆ [ n ] ♭ are given by the cosimplicial diagram ∆ in s S et and the map ∆ [ ] ♭ → ∆ [ ] ♯ is the evident inclusion map.We can depict the marked cosimplicial object τ in s S et + as ∆ [ ] ♭ ∆ [ ] ♭ ∆ [ ] ♭ · · · ∆ [ ] .Using τ we can now define a functor t + : ∆ + × ∆ → s S et + by t + ([ n ] , [ m ]) = τ ( n ) × ∆ [ n ] . We can extend this functor to an adjunction. Definition 2.15.
Let Fun (( ∆ + ) op × ∆ op , S et ) s S et +( t + ) ! ⊥ ( t + ) ! be the adjunction given by left Kan extension along t + .The right hand side of the adjunction already takes value in marked simplicial sets, so we only need toprove that the right adjoint takes value in marked simplicial spaces as well. Lemma 2.16.
The adjunction (( t + ) ! , ( t + ) ! ) restricts to an adjunctions S + s S et +( t + ) ! ⊥ ( t + ) ! Proof.
Let ( X , A ) be a marked simplicial set. Then ( t + ) ! ( X , A ) nl = Hom s S et + ( τ ( n ) × ∆ [ l ] , ( X , A )) ∼ = Hom s S et + ( τ ( n ) , ( X , A ) ∆ [ l ] ) ,where the isomorphism follows from the fact that s S et + is Cartesian closed (Remark 2.5).In order to show that ( t + ) ! ( X , A ) is a marked simplicial space we have to prove that mapHom s S et + ( τ ( + ) × ∆ [ l ] , ( X , A )) → Hom s S et + ( τ ( ) × ∆ [ l ] , ( X , A )) is an injection for all l ≥
0. By direct computation this map is given byHom s S et + ( ∆ [ ] ♯ × ∆ [ l ] , ( X , A )) → Hom s S et + ( ∆ [ ] ♭ × ∆ [ l ] , ( X , A )) . By the isomorphism above, this is isomorphic toHom s S et + ( ∆ [ ] ♯ , ( X , A ) ∆ [ l ] ) → Hom s S et + ( ∆ [ ] ♭ , ( X , A ) ∆ [ l ] ) .Finally, by Remark 2.8, the right hand side is the set of 1-simplices in ( X , A ) ∆ [ l ] , whereas the left hand sideare the marked ones. Hence, this map is an injection by assumption. (cid:3) Remark . Recall that t : ∆ × ∆ → s S et was defined as t ([ n ] , [ m ]) = ∆ [ n ] × J [ m ] (Theorem 1.22). Hence ourfirst guess for a generalization to marked simplicial spaces might have been t ′ ([ n ] , [ m ]) = τ ([ n ]) × J [ m ] ♭ .This functor would in fact give us an adjunction between marked simplicial spaces and marked simpli-cial sets and we could use that to construct Quillen equivalences. However, in 2.23 we observe that themodel structure of interest on s S et + , the marked Joyal structure, is in fact a simplicial model structure, andthat the simplicial enrichment makes (( t + ) ! , ( t + ) ! ) into a simplicial Quillen equivalence Theorem 2.26. Ifwe used (( t ′ ) ! , ( t ′ ) ! ) instead we would not get a simplicial Quillen equivalence.On the other hand, we do in fact use the fact that (( t + ) ! , ( t + ) ! ) is simplicial, particularly in 2.26 and 2.44,to construct our desired model structures on marked simplicial spaces (we need it to apply Theorem 1.9).Hence we chosen t + as our generalization t .Finally, we note that even without any particular application in mind, knowing that a Quillen equiva-lence is simplicial brings certain benefits, giving additional justification for our choice of generalization. Remark . Notice the adjunctions (( t + ) ! , ( t + ) ! ) and ( t ! , t ! ) do not give us a commutative square. Thusour choice of prioritizing a simplicial Quillen equivalence (as explained in Remark 2.17) comes at the priceof not commuting with the original adjunction ( t ! , t ! ) .We now want to move on and prove that the adjunctions (( p + ) ∗ , ( i + ) ∗ ) and (( t + ) ! , ( t + ) ! ) do in fact giveus a Quillen equivalence, similar to the unmarked counter-parts.For that we first have to construct appropriate model structures, which we will do in two steps:(1) First we define a model structure s S + which is transferred from a model structure on s S and doesnot take the markings into account (Proposition 2.19).(2) Then we localize this model structure such that the fibrancy condition depends on an appropriatechoice of marking (Theorem 2.26). Proposition 2.19.
Let L be a set of cofibrations of simplicial spaces. There is a unique combinatorial simplicialleft proper model structure on the category of marked simplicial spaces, called the unmarked L -localized modelstructure and denoted by ( s S + ) un + Ree L , with the following specifications:F A map ( X , A ) → ( Y , B ) is a fibration if the underlying map of simplicial spaces X → Y is a fibration in the L localized Reedy model structure.W A map ( X , A ) → ( Y , B ) is a weak equivalence if the underlying map of simplicial spaces X → Y is a weakequivalence in the L -localized Reedy model structure.C A map of marked simplicial spaces is a cofibration if it has the left lifting property with respect to maps thatare simultaneously fibrations and weak equivalences.Moreover, the adjunction s S Ree L ( s S + ) un + Ree L ( − ) ♭ ⊥ Unm .is a Quillen equivalence. Here the left hand side has the L -localized model structure (Theorem 1.7) and the right sidethe unmarked L -localized model structure. UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 21
Proof.
In order to construct a model structure on s S + we want to prove that we can use the adjunction(2.20) s S Ree L s S +( − ) ♭ ⊥ Unm to induce the L -localized Reedy model structure via the right adjoint Unm. This would immediately giveus most of the desired results, up to including that the adjunction is a Quillen adjunction, leaving us onlywith the task of showing it is a Quillen equivalence.We will hence check the required conditions of right-induced model structures (Theorem 1.8): • Unm commutes with filtered colimits : This follows from the fact that Unm is a left adjoint (Remark 2.12). • s S + is simplicial : Also proven in Remark 2.12. • There is a functorial fibrant replacement functor in s S + : Let R be a fibrant replacement functor in s S Ree L .Then the functor s S + Unm −−−→ s S R −→ s S ( − ) ♯ −−−→ s S is in fact a fibrant replacement functor. Indeed for a given marked simplicial space ( X , A ) we haveequivalences: ( X , A ) ≃ −→ ( X , X ) ≃ −→ ( RX , RX ) .We are now left with proving that the Quillen adjunction (2.20) is in fact a Quillen equivalence. Theright adjoint reflects weak equivalences by definition and so it suffices to prove the derived unit map is anequivalence for every fibrant simplicial space, however, that is just the identity. Hence we are done. (cid:3) We can now use this result for particular interesting instances.
Corollary 2.21. If L is empty, then we get the unmarked Reedy model structure, ( s S + ) un + Ree , on marked simplicialspaces.
Corollary 2.22.
If we take L = S eg ∪ C omp (the Segal maps and completeness maps as defined in Definition 1.13),then we get the unmarked complete Segal space model structure, denoted ( s S + ) un + CSS . The reason the model structure is called “unmarked” is that the fibrancy condition is independent ofthe markings. This has very problematic implications, such as the fact that even some common markedsimplicial spaces are not cofibrant anymore. For example the diagram F ( ) ♭ F ( ) ♯ F ( ) ♯ ∄ id has no lift, although right hand map is a trivial fibration in the L -localized marked Reedy model structure.Hence F ( ) ♯ is not cofibrant.We thus want modify the model structure, by restricting the fibrations and trivial fibrations and makingit dependent on the markings. In order to do that we want to compare the unmarked CSS model structurewith the marked Joyal model structure . Theorem 2.23. [Lur09a, Proposition 3.1.3.7, Proposition 3.1.4.1, Corollary 3.1.4.4, Proposition 3.1.5.3]
Thereis a unique combinatorial, simplicial, left proper model structure on s S et + , called the marked Joyal model structureand denoted s S et Joy + , with the following specifications:(1) A map of marked simplicial sets f : ( S , A ) → ( T , B ) is a cofibration if the underlying map of simplicial setsS → T is a monomorphism. (2) A map of marked simplicial sets f : ( S , A ) → ( T , B ) is a weak equivalence if the underlying map of simplicialsets S → T is an equivalence in the Joyal model structure.(3) A marked simplicial set ( X , A ) is fibrant if X is a quasi-category and A is the set of equivalences in X.(4) The simplicial enrichment is given by the simplicial set Map s S et + ( S , T ) = Hom s S et + ( S × ∆ [ n ] ♯ , T ) Finally, the adjunction s S et Joy ( s S et + ) Joy + ( − ) ♭ ⊥ Unm is a Quillen equivalence between the Joyal model structure and the marked Joyal model structure.Remark . This model structure is originally defined in [Lur09a] as a special case of the Cartesian modelstructure for marked simplicial sets (which we will review in Theorem 2.37) and hence does not have itsown separate name there.We want to use the marked Joyal model structure to adjust the unmarked CSS model structure.
Lemma 2.25.
The adjunction ( s S + ) un + CSS ( s S et + ) Joy + ( t + ) ! ⊥ ( t + ) ! is a simplicial Quillen adjunction between the unmarked CSS model structure and the marked Joyal model structure.Proof. First, the adjunction is in fact simplicial. Indeed, we have ( t + ) ! ( ∆ [ n ] ♭ ) = ∆ [ n ] .As all monomorphisms in ( s S et + ) Joy + are cofibrations, the left adjoint ( t + ) ! indeed preserves cofibra-tions. Hence, in order to prove we have a Quillen adjunction it suffices to prove ( t + ) ! preserves fibrantobjects.Let ( S , S hoequiv ) be a fibrant object (here S is a quasi-category). Then, we haveHom s S et + ( ∆ [ ] ♯ , ( S , S hoequiv )) ∼ = S hoequiv ∼ = Hom s S et ( J [ ] , S ) ∼ = Hom s S et ( J [ ] ♭ , ( S , S hoequiv )) Applying this bijection to the the definition of t ! (Theorem 1.22) implies that ( t + ) ! ( S , S hoequiv ) = ( t ! ( S ) , t ! ( S ) hoequiv ) ,which is indeed fibrant as t ! ( S ) is a complete space (again by Theorem 1.22). (cid:3) Using this Quillen adjunction we can finally construct a new model structure on s S + , in which the fibrantobjects depends on the markings! Theorem 2.26.
There exists a unique simplicial, cofibrantly generated, left proper model structure on marked sim-plicial spaces, denoted ( s S + ) CSS + and called the marked CSS model structure, characterized by(1) Fibrant objects are marked simplicial spaces of the form ( W , W hoequiv ) , where W is a complete Segal spaceand W hoequiv is the subspace of weak equivalences.(2) A map f : ( X , A ) → ( Y , B ) is a cofibration if the map of underlying simplicial spaces X → Y is a monomor-phism.(3) A map f : ( X , A ) → ( Y , B ) is a weak equivalence if for every fibrant object ( W , W hoequiv ) the induced map Map s S + (( Y , B ) , ( W , W hoequiv )) → Map s S + (( X , A ) , ( W , W hoequiv )) is a Kan equivalence. UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 23 (4) A map between fibrant objects ( W , W hoequiv ) → ( V , V hoequiv ) is a weak equivalence (fibration) if and only ifit is a weak equivalence (fibration) in the unmarked CSS model structure.(5) The adjunction (2.27) s S + s S et +( t + ) ! ⊥ ( t + ) ! is a simplicial Quillen equivalence of simplicial model structures.Proof. By Lemma 2.25, we have a simplicial Quillen adjunction ( s S + ) un + CSS ( s S et + ) Joy + ( t + ) ! ⊥ ( t + ) ! .Hence, by applying Theorem 1.9, we get a new model structure on s S + , which we call the marked CSS modelstructure . We want to prove that this model structure satisfies all the conditions stated above. We will startby proving that the adjunction (( t + ) ! , ( t + ) ! ) is a Quillen equivalence.By Theorem 1.9, it suffices to show that the derived right Quillen functor, ( t + ) ! , is fully faithful. Fix twofibrant objects ( S , S hoequiv ) , ( T , T hoequiv ) in ( s S et + ) Joy + . We want to proveMap s S et + (( S , S hoequiv ) , ( T , T hoequiv )) → Map s S + (( t + ) ! ( S , S hoequiv ) , ( t + ) ! ( T , T hoequiv )) is a Kan equivalence. This will require several computations.First, we observe that a ∆ [ n ] ♯ → ( S , S hoequiv ) sends all edges to weak equivalences, and so we havebijection(2.28) Hom s S et + (( S , S hoequiv ) , ( T , T hoequiv )) ∼ = Hom s S et ( S , T ) .Moreover, we observe that any map of simplicial sets S → T will automatically send S hoequiv to T hoequiv .Giving us the bijection(2.29) Hom s S et + ( ∆ [ n ] ♯ , ( S , S hoequiv )) ∼ = Hom s S et ( J [ n ] , S ) .Combining these we haveMap s S et + (( S , S hoequiv ) , ( T , T hoequiv )) n = Hom s S et + (( S , S hoequiv ) × ∆ [ n ] ♯ , ( T , T hoequiv )) ( ) ∼ = Hom s S et + (( S , S hoequiv ) × ( J [ n ] , J [ n ] hoequiv ) , ( T , T hoequiv )) ( ) ∼ = Hom s S et ( S × J [ n ] , T ) = t ! ( T S ) n .Next, by the argument in the proof of Lemma 2.25 we have(2.30) ( t + ) ! ( S , S h oequiv ) = ( t ! S , t ! S hoequiv ) .Moreover, using the same argument as above, we have a bijection(2.31) Hom s S + (( S , S hoequiv ) , ( T , T hoequiv )) ∼ = Hom s S ( S , T ) .Combining these we haveMap s S + (( t + ) ! ( S , S hoequiv ) , ( t + ) ! ( T , T hoequiv )) n = Hom s S + (( t + ) ! ( S , S hoequiv ) × ∆ [ n ] , ( t + ) ! ( T , T hoequiv )) ( ) ∼ = Hom s S + (( t ! S , t ! S hoequiv ) × ∆ [ n ] , ( t ! T , t ! T hoequiv )) ( ) ∼ = Hom s S ( t ! S × ∆ [ n ] , t ! T ) = (( t ! T ) t ! S ) n . Thus, in order to prove that the derived functor ( t + ) ! is fully faithful it suffices to prove that the map t ! ( T S ) → ( t ! T t ! S ) is a Kan equivalence. However, this immediately follows from that fact that the map t ! ( T S ) → t ! ( T ) t ! ( S ) isa CSS equivalence (Remark 1.24). This proves that (( t + ) ! , ( t + ) ! ) is a Quillen equivalence.We now move on to prove that cofibrations are monomorphisms. By Theorem 1.9, a map is a cofibrationif and only its image under ( t + ) ! is a cofibration. However, the cofibrations in the marked Joyal modelstructure are just the monomorphisms and so the result follows.Next, we characterize the fibrant objects. By definition of the localization model structure, an object ( W , A ) is fibrant if it is fibrant in the unmarked CSS model structure, meaning W is a complete Segal space,and it lies in the essential image of the right adjoint ( t + ) ! , hence A = W hoequiv .Next notice the marked CSS model structure is simplicial and fibrant objects are of the form ( W , W hoequiv ) and so a map ( Y , A ) → ( Z , B ) is an equivalence if and only ifMap s S + (( Z , B ) , ( W , W hoequiv )) → Map s S + (( Y , A ) , ( W , W hoequiv )) is a Kan equivalence for every complete Segal space W .Finally, the model structure is given as a localization model structure of the unmarked CSS model struc-ture. Hence, fibrations (weak equivalences) between fibrant objects are given by fibrations (weak equiva-lences) in the unmarked CSS model structure. (cid:3) We can now use our results about the marked CSS model structure to observe various interesting Quillenequivalences.
Theorem 2.32.
The following diagram s S et Joy s S CSS ( s S et + ) Joy + ( s S + ) CSS + p ∗ ⊥ ( − ) ♭ ⊥ ( − ) ♭ ⊥ i ∗ ( p + ) ∗ ⊥ Unm Unm ( i + ) ∗ is a commutative diagram of Quillen equivalences. Here the top row has the Joyal and CSS model structure. Thebottom row has the marked Joyal and marked CSS model structure.Proof. First we show the diagram commutes. It suffices to observe that the left adjoints commute. Moreover,left adjoints commute with colimits and so it suffices to check on generators. However, both sides of thesquare map ∆ [ n ] to F ( n ) ♭ .We now move on to prove all four are Quillen equivalences. By Theorem 1.23, we already know that ( p ∗ , i ∗ ) is a Quillen equivalence. Similarly, by Theorem 2.23, we know that the left hand (( − ) ♭ , Unm ) is aQuillen equivalence. Hence we have to focus on the other two.First we show that (( p + ) ∗ , ( i + ) ∗ ) is a Quillen adjunction. Evidently ( p + ) ∗ preserves cofibrations, whichare just the monomorphisms. Hence it suffices to prove that ( i + ) ∗ preserves fibrant objects and fibrationsbetween fibrant objects. However, fibrant objects are of the form ( W , W hoequiv ) , where W is a completeSegal space. By direct computation, ( i + ) ∗ ( W , W hoequiv ) = ( i ∗ ( W ) , i ∗ ( W ) ) , where i ∗ ( W ) is a quasi-category(Theorem 1.23). Moreover, fibrations between fibrant objects are just unmarked CSS fibrations (), which are,again by Theorem 1.23, taken to Joyal fibrations. UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 25
Next, we prove that (( p + ) ∗ , ( i + ) ∗ ) is a Quillen equivalence. We can extend our adjunction as follows: ( s S et + ) Joy + ( s S + ) CSS + ( s S et + ) Joy + ( p + ) ∗ ⊥ ( t + ) ! ⊥ ( i + ) ∗ ( t + ) ! .By Theorem 2.26, (( t + ) ! , ( t + ) ! ) is a Quillen equivalence. Moreover, the composition ( t + ) ! ( p + ) ∗ is just theidentity and so is by default a Quillen equivalence. Hence, by 2-out-of-3, (( p + ) ∗ , ( i + ) ∗ ) is a Quillen equiv-alence.Next, we show (( − ) ♭ , Unm ) is a Quillen adjunction. ( − ) ♭ preserves cofibrations as they are just themonomorphisms. So,we only have to prove that Unm preserves fibrant objects and fibrations betweenfibrant objects. Similar to above, fibrant objects are just ( W , W hoequiv ) , where W is a complete Segal spaceand fibrations are just Reedy fibrations, both of which are preserves by Unm.Finally, we prove that (( − ) ♭ , Unm ) is a Quillen equivalence. However, this follows immediately fromthe fact that the other three adjunctions are Quillen equivalences and 2-out-of-3. (cid:3) Combining Theorem 2.26 and Theorem 2.32 with Proposition 1.10 gives us following useful corollary.
Corollary 2.33.
Let ( X , A ) be a marked simplicial space and ( S , B ) be a marked simplicial set. We have followingQuillen equivalences ( s S + ) CSS + / X ( s S et + ) Joy + / ( t + ) ! ( X )( t + ) ! ⊥ u ∗ ( t + ) ! ( s S et + ) Joy + / S ( s S + ) CSS + / ( p + ) ∗ S ( p + ) ∗ ⊥ u ∗ ( i + ) ∗ .Moreover, the adjunction (( t + ) ! , u ∗ ( t + ) ! ) is a simplicial Quillen equivalence. Taking X to be the final object, the result above implies that we have constructed another model structurefor ( ∞ , 1 ) -categories on the category of marked simplicial spaces. Remark . We can summarize the results of this subsection with the following diagram of Quillen equiv-alences ( s S + ) un + CSS s S CSS ( s S et + ) Joy + ( s S + ) CSS + Unm ( t + ) ! ⊥ id ⊥ ( − ) ♭ ⊥ ( − ) ♭ ⊥ ( t + ) ! ( t + ) ! Unm ( t + ) ! ⊥ id The CSS model structure on the left hand side, s S CSS , and the marked Joyal model structure on the righthand side, ( s S et + ) Joy + , already existed and we build a bridge from one to the other by defining the markedCSS model structure, ( s S + ) CSS + .So, what is the role of the unmarked CSS model structure, ( s S + ) un + CSS ? If we wanted to directly con-struct a model structure on s S + using the adjunction (( t + ) ! , ( t + ) ! ) , we would have to left induce our modelstructure from the marked Joyal model structure. However, left-induced model structures are quite difficultto construct [HKRS17]. Hence, we used the unmarked CSS model structure, ( s S + ) un + CSS , as an intermediate step to avoid suchdifficulties. We can easily right induce this model structure from the CSS model structure, s S CSS (as we didin Proposition 2.19). Now, as soon as we have any appropriate model structure on s S + , we can then usethe theory of Bousfield localizations to get the desired marked CSS model structure, ( s S + ) CSS + . Thus, wecan think of the unmarked CSS model structure as an intermediate step that allows us to easily define themarked CSS model structure.We will use these adjunctions in the next subsection to define and study the Cartesian model structureof marked simplicial spaces. In this section we define Cartesian fibrationsof simplicial spaces and prove that we can define a model structure on marked simplicial spaces over asimplicial space such that the fibrant objects are precisely the Cartesian fibrations. We also show that thismodel structure is Quillen equivalent to the Cartesian model structure on marked simplicial sets via bothadjunctions (( t + ) ! , ( t + ) ! ) , (( p + ) ∗ , ( i + ) ∗ ) .First, we review the definition of Cartesian fibrations and Cartesian model structure of simplicial sets, asdiscussed in [Lur09a]. Definition 2.35. [Lur09a, Definition 2.4.1.1.] Let p : T → S be a map of simplicial sets. An edge f : x → y in T is p -Cartesian if T / f ։ S / p ( f ) × S / p ( y ) T / y is a trivial Kan fibration. Here, T / y is defined in [Lur09a, Proposition 1.2.9.2]. Definition 2.36.
An inner fibration p : T → S is a Cartesian fibration , if for every edge f : x → y in S and lift˜ y , there exist a p -Cartesian lift ˜ f : ˜ x → ˜ y .Cartesian fibrations are fibrant objects in a model structure on marked simplicial sets. Theorem 2.37. [Lur09a, Proposition 3.1.3.7, Proposition 3.1.4.1, Corollary 3.1.4.4, Proposition 3.1.5.3]
Let Sbe a simplicial set. There exists a unique left proper, combinatorial, simplicial model structure on s S et + / S which maybe described as follows: C A map is a cofibration if the map of underlying simplicial sets is a monomorphism. F Fibrant objects are of the form ( T , T Cart ) → S , where T → S is a Cartesian fibration and T
Cart the set ofCartesian edges. W A morphism g is an equivalence if for every Cartesian fibration T → S the induced map
Map / S ( g , T ) is aKan equivalence.The simplicial enrichment is given by Map / S ( T , U ) n = Hom / S ( T × ∆ [ n ] , U ) . Finally, for every simplicial set S the adjunction ( s S et / S ) Joy + ( s S et + / S ) Cart id ⊥ id is a Quillen adjunction between the Joyal model structure and the Cartesian model structure.Notation . In [Lur09a] a map of the form ( T , T Cart ) → S where T → S is a Cartesian fibration is denotedby T ♮ over S , however, we will not use this notation. Example 2.39. If S = ∆ [ ] then the Cartesian model structure simply recovers the marked Joyal modelstructure introduced in Theorem 2.23. UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 27
Our goal is to define Cartesian fibrations of marked simplicial spaces and prove it comes with a modelstructure that is equivalent to the Cartesian model structure on marked simplicial sets.
Definition 2.40.
A CSS fibration of simplicial spaces Y → X is a Cartesian fibration if i ∗ Y → i ∗ X is aCartesian fibration of simplicial sets.We now want to give an internal description of Cartesian fibrations of simplicial spaces, that does notrely on simplicial sets. This requires some category theory of simplicial spaces, as studied in [Ras17b].Let X be a complete Segal space and f a morphism. Then the the complete Segal space of cones X / f [Ras17b,Definition 5.21] is defined as X / f = ( X F ( ) ) F ( ) × X F ( ) × X F ( ) X × F ( ) .Generalizing from there, if X is an arbitrary simplicial space, then we can choose a complete Segal spacefibrant replacement X → ˆ X and define X / f = X × ˆ X ˆ X / f , using the fact that cocones are invariant un-der equivalences of complete Segal spaces [Ras17b, Theorem 5.1]. Note that, up to Reedy equivalence ofsimplicial spaces, this definition is independent of the choice of ˆ X . Definition 2.41.
Let p : V → W be a CSS fibration of simplicial spaces. A morphism f : x → y in V is called p -Cartesian if the commutative square V / f V / y W / p ( f ) W / p ( y ) is a homotopy pullback square of simplicial spaces.We can now use this to give an internal characterization of Cartesian fibrations. Lemma 2.42.
Let p : V → W be CSS fibration. Then p is a Cartesian fibration if and only if for every morphismf : x → y in W and lift ˆ y in V of y, there exists a p-Cartesian morphism lift ˆ f of f (meaning p ( ˆ f ) = f ).Proof. Before we start the proof, we recall that i ∗ ( W ) = W , which means W and i ∗ W have the same setof edges.First, by Theorem 1.23, if p : V → W is a CSS fibration then i ∗ ( p ) : i ∗ ( V ) → i ∗ ( W ) is a fibration in theJoyal model structure. Thus, by Definition 2.36, i ∗ ( p ) : i ∗ ( V ) → i ∗ ( W ) is a Cartesian fibration of simplicialsets if and only if every edge f : x → y in i ∗ ( W ) with given lift ˆ y has a i ∗ ( p ) -Cartesian lift.Hence the statement follows immediately from proving that an edge f in V is p -Cartesian if and only if f in i ∗ ( V ) is i ∗ ( p ) -Cartesian.However, this follows from the fact that i ∗ is a right Quillen functor of a Quillen equivalence and so V / f V / y i ∗ ( V ) / f i ∗ ( V ) / y W / p ( f ) W / p ( y ) i ∗ ( W ) / p ( f ) i ∗ ( W ) / p ( y ) the right square is a homotopy pullback square if and only if the left hand square is a homotopy pullbacksquare. (cid:3) We now want to move our study of Cartesian fibrations into the marked world and so need an appro-priate definition.
Definition 2.43.
Let X be a simplicial space. We say a map of marked simplicial space ( Y , Y Cart ) → X is a Cartesian fibration if Y → X is a Cartesian fibration and Y Cart is the sub-space of Cartesian edges.We are now ready to show that Cartesian fibrations (Definition 2.43) are fibrant objects in a model struc-ture.
Theorem 2.44.
Let X be a simplicial space. There is a unique combinatorial, left proper, simplicial model structureon ( s S + ) / X , called the Cartesian model structure and denoted ( s S / X ) Cart , such that(1) Cofibrations are monomorphisms over X .(2) Fibrant objects are Cartesian fibrations ( Y , Y Cart ) → X .(3) A map is a weak equivalence if for every Cartesian fibration ( W , D ) → X ♯ the induced map Map / X ♯ (( Z , C ) , ( W , D )) → Map / X ♯ (( Y , B ) , ( W , D )) is a Kan equivalence.(4) A map between Cartesian fibrations ( W , W Cart ) → ( V , V Cart )) over X ♯ is a weak equivalence (fibration) ifand only if it is a weak equivalence (fibration) in the marked CSS model structure.(5) The adjunction (( s S + ) / X ) Cart (( s S et + ) / ( t + ) ! X ) Cart ( t + ) ! ⊥ u ∗ ( t + ) ! is a simplicial Quillen equivalence.Proof. Combining Corollary 2.33 and Theorem 2.37, we have simplicial Quillen adjunctions(2.45) (( s S + ) / X ) CSS + (( s S et + ) / ( t + ) ! X ) Joy + (( s S et + ) / ( t + ) ! X ) Cart ( t + ) ! ⊥ id ⊥ u ∗ ( t + ) ! id between combinatorial simplicial model structures, where the right adjoint is fully faithful (by Corollary 2.33).Hence, by Theorem 1.9, we can define a new model structure on s S + / X , which we call the Cartesian modelstructure and that makes the adjunction (( t + ) ! , u ∗ ( t + ) ! ) into a simplicial Quillen equivalence.Next, combining the facts that ( t + ) ! reflects cofibrations (Theorem 1.9) and that cofibrations in (( s S et + ) / t ! X ) Cart are monomorphisms implies that cofibrations in (( s S + ) / X ) Cart are also monomorphisms.Next, we characterize the fibrant objects, which are precisely the fibrant objects in ( s S + / X ) CSS + thatare in the essential image of u ∗ ( t + ) ! . By Theorem 2.32, if a map ( Y , B ) → X ♯ is a marked CSS fibra-tion, then Y → X is a CSS fibration of simplicial spaces. On the other hand, ( Y , B ) is in the image if ( Y , B ) = ( t + ) ∗ ( C , C Cart ) = ( t ! C , ( t ! C ) Cart ) , which implies that B = Y Cart . Hence, the fibrant objects areprecisely the Cartesian fibrations over X .The characterization of weak equivalences follows from the fact that the model structure is simplicialand that the fibrant objects are given by Cartesian fibrations of marked simplicial spaces.Finally, the model structure is given as a localization model structure of the marked CSS model structure.Hence, fibrations (weak equivalences) between fibrant objects are given by fibrations (weak equivalences)in the marked CSS model structure. (cid:3) We constructed the Cartesian model structure on ( s S + ) / X as a localization model structure. Hence, weget following result. Corollary 2.46.
Let X be a simplicial space. Then the adjunction
UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 29 (( s S + ) / X ) CSS + (( s S + ) / X ) Cart id ⊥ id is a Quillen adjunction. We end this section by giving a second Quillen equivalence between the Cartesian model structures.
Theorem 2.47.
Let S be a simplicial set. The adjunction (( s S et + ) / S ) Cart (( s S + ) / ( p + ) ∗ S ) Cart ( p + ) ∗ ⊥ u ∗ ( i + ) ∗ is a Quillen equivalence between Cartesian model structures.Proof. In order to prove it is a Quillen adjunction it suffices to show the left adjoint preserves cofibrationsand the right adjoint preserves fibrant objects and trivial fibrations between fibrant objects. Evidently, ( p + ) ∗ preserves cofibrations as they are just monomorphisms.Moreover, ( i + ) ∗ preserves trivial fibrations between fibrant objects. Indeed, by Theorem 2.44, weakequivalences between fibrant objects are just marked CSS equivalences, which, by Theorem 2.32, are mappedto marked Joyal equivalences, which, by Theorem 2.37, are Cartesian equivalences.We are left with proving that ( i + ) ∗ preserves fibrant object. However, by Theorem 2.44, fibrant ob-jects are of the form ( Y , Y Cart ) → X ♯ , where Y → X is a Cartesian fibration. By direct computation, ( i + ) ∗ (( Y , Y Cart ) → X ♯ ) = ( i ∗ Y , i ∗ Y Cart ) → i ∗ X ♯ , which is fibrant in the Cartesian model structure in ( s S et + / i ∗ X ♯ ) Cart (Theorem 2.37).We move on to prove that the adjunction is a Quillen equivalence. We can extend the adjunction aboveto the following diagram (( s S et + ) / S ) Cart (( s S + ) / ( p + ) ∗ S ) Cart (( s S et + ) / S ) Cart ( p + ) ∗ ⊥ ( t + ) ! ⊥ u ∗ ( i + ) ∗ u ∗ ( t + ) ! .The second adjunction is a Quillen equivalence by the previous theorem. The composition is the identityand so also a Quillen equivalence. Hence the left hand Quillen adjunction is a Quillen equivalence, by2-out-of-3. (cid:3) E QUIVALENCE OF C ARTESIAN M ODEL S TRUCTURES : M
ODEL D EPENDENT AND I NDEPENDENT
Until this point we have studied many different model structures, that we all denoted by
Cartesian modelstructure : for marked simplicial sets, marked simplicial spaces, bisimplicial sets and bisimplicial spaces. Weare finally in a position where we can show that all these model structures are in fact Quillen equivalent viaa zig-zag of Quillen equivalence and hence we are justified in giving all of them the same name.Finally, in the last subsection, we show how this immediately gives us an equivalence of underlyingquasi-categories, where we can replace a zig-zag of equivalences with direct adjoint equivalences.
In this section we want to prove that we havefollowing diagram of Quillen equivalences (the direction of the arrows are the direction of the right ad-joints). Here an arrow marks a right Quillen functor of a Quillen equivalence, X is a simplicial space and S a simplicial set. Marked Simplicial Functorial Bisimplicial ( s S et + / t ! X ) Cart
Fun ( C [ t ! X ] , ( s S et + ) Joy + ) proj ( s S + / X ) Cart
Fun ( C [ t ! X ] , ( s S + ) CSS + ) proj ( ss S / X ) Cart
Fun ( C [ t ! X ] , s S CSS ) proj ( ss S et / t ! X ) Cart ( s S et + / S ) Cart
Fun ( C [ S ] , ( s S et + ) Joy + ) proj ( s S + / p ∗ S ) Cart
Fun ( C [ S ] , ( s S + ) CSS + ) proj ( ss S / p ∗ S ) Cart
Fun ( C [ S ] , s S CSS ) proj ( ss S et / S ) Cart (( t + ) ! , u ∗ ( t + ) ! ) ( St + t ! X ,Un + t ! X ) ((( t + ) ! ) ∗ , (( t + ) ! ) ∗ )((( − ) ♭ ) ∗ , ( Unm ) ∗ ) ( sSt t ! X ,sUn t ! X ) ( st ! , su ∗ t ! )( St + S ,Un + S ) ((( t + ) ! ) ∗ , (( t + ) ! ) ∗ )(( p + ) ∗ , u ∗ ( i + ) ∗ ) ((( − ) ♭ ) ∗ , ( Unm ) ∗ ) ( sp ∗ , su ∗ i ∗ )( sSt S ,sUn S ) • (( t + ) ! , u ∗ ( t + ) ! ) : This is proven in Theorem 2.44. • (( p + ) ∗ , u ∗ ( i + ) ∗ ) : This is proven in Theorem 2.47. • ( st ! , su ∗ t ! ) : This is proven in Theorem 1.35. • ( sp ∗ , su ∗ i ∗ ) : This is proven in Theorem 1.35. • ((( t + ) ! ) ∗ , (( t + ) ! ) ∗ ) This follows from applying Lemma 1.6 to the Quillen equivalence ((( t + ) ! ) ∗ , (( t + ) ! ) ∗ ) (Theorem 2.26). • ((( − ) ♭ ) ∗ , ( Unm ) ∗ ) : This follows from applying Lemma 1.6 to the Quillen equivalence ((( − ) ♭ ) ∗ , ( Unm ) ∗ ) (Theorem 2.32). • ( sSt S , sUn S ) : This follows from applying Theorem 1.21 to the Quillen equivalence ( St S , Un S ) [Lur09a,Theorem 2.2.1.2]. • ( St + S , Un + S ) : This is the statement of [Lur09a, Theorem 3.2.0.1]. The work here focused on model categorical formalism, but inthis subsection we want to give short analysis of the quasi-categorical implications of this result. The modelstructures from Subsection 3.1 are all simplicial. So, we can apply the simplicial nerve N ([Lur09a, 1.1.5])and the the fact that Quillen equivalences of simplicial model categories gives us adjoint equivalences of UASI-CATEGORIES VS. SEGAL SPACES: CARTESIAN EDITION 31 quasi-categories [Lur09a, Proposition 5.2.4.6], to get a diagram of equivalences of quasi-categories (here weonly focus on the case over a simplicial space X ): N (cid:16) ( s S et + / t ! X ) Cart (cid:17) N (cid:0) Fun ( C [ t ! X ] , ( s S et + ) Cart ) proj (cid:1) N (cid:16) ( s S + / X ) Cart (cid:17) N (cid:16) Fun ( C [ t ! X ] , ( s S + ) CSS + ) proj (cid:17) N (cid:0) ( ss S / X ) Cart (cid:1) N (cid:0) Fun ( C [ t ! X ] , s S CSS ) proj (cid:1) N (cid:0) ( ss S et / t ! X ) Cart (cid:1) (( t + ) ! , u ∗ ( t + ) ! ) ( St + t ! X ,Un + t ! X ) ((( t + ) ! ) ∗ , (( t + ) ! ) ∗ )((( − ) ♭ ) ∗ , ( Unm ) ∗ ) ( sSt t ! X ,sUn t ! X ) ( st ! , su ∗ t ! ) By [RV21, 2.1.12], whenever we have an adjoint equivalence of quasi-categories, both functors are right andleft adjoints. Thus we can ignore the direction of the arrows. This is clearly not true for Quillen equivalencesof model categories, as functors (even equivalences) are usually not right and left Quillen.R
EFERENCES[AF20] David Ayala and John Francis. Fibrations of ∞ -categories. High. Struct. , 4(1):168–265, 2020. (On p. 2)[Ara14] Dimitri Ara. Higher quasi-categories vs higher Rezk spaces.
J. K-Theory , 14(3):701–749, 2014. (On p. 5)[Bar05] Clark Barwick. (infinity, n)-Cat as a closed model category . ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University ofPennsylvania. (On p. 5)[BdB18] Pedro Boavida de Brito. Segal objects and the Grothendieck construction. In
An alpine bouquet of algebraic topology , volume708 of
Contemp. Math. , pages 19–44. Amer. Math. Soc., Providence, RI, 2018. (On pp. 1 and 3)[Ber10] Julia E. Bergner. A survey of ( ∞ , 1 ) -categories. In Towards higher categories , volume 152 of
IMA Vol. Math. Appl. , pages 69–83.Springer, New York, 2010. (On p. 1)[Ber20] Julia E Bergner. A survey of models for ( ∞ , n ) -categories. Handbook of Homotopy Theory, edited by Haynes Miller, Chapman &Hall/CRC , pages 263–295, 2020. (On p. 5)[BR13] Julia E. Bergner and Charles Rezk. Comparison of models for ( ∞ , n ) -categories, I. Geom. Topol. , 17(4):2163–2202, 2013. (Onp. 5)[BR20] Julia E. Bergner and Charles Rezk. Comparison of models for ( ∞ , n ) -categories, II. J. Topol. , 13(4):1554–1581, 2020. (On p. 5)[BV73] J. M. Boardman and R. M. Vogt.
Homotopy invariant algebraic structures on topological spaces . Lecture Notes in Mathematics,Vol. 347. Springer-Verlag, Berlin-New York, 1973. (On pp. 2 and 10)[CKM20] Timothy Campion, Krzysztof Kapulkin, and Yuki Maehara. Comical sets: A cubical model for ( ∞ , n ) -categories. arXivpreprint , 2020. arXiv:2005.07603v2. (On p. 5)[Con07] Brian Conrad. Arithmetic moduli of generalized elliptic curves. J. Inst. Math. Jussieu , 6(2):209–278, 2007. (On p. 1)[CS19] Damien Calaque and Claudia Scheimbauer. A note on the ( ∞ , n ) -category of cobordisms. Algebr. Geom. Topol. , 19(2):533–655, 2019. (On p. 5)[GJ99] Paul G. Goerss and John F. Jardine.
Simplicial homotopy theory , volume 174 of
Progress in Mathematics . Birkh¨auser Verlag,Basel, 1999. (On p. 7)[GR17a] Dennis Gaitsgory and Nick Rozenblyum.
A study in derived algebraic geometry. Vol. I. Correspondences and duality , volume221 of
Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2017. (On p. 5)[GR17b] Dennis Gaitsgory and Nick Rozenblyum.
A study in derived algebraic geometry. Vol. II. Deformations, Lie theory and formalgeometry , volume 221 of
Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2017. (Onp. 5)[gro03]
Revˆetements ´etales et groupe fondamental (SGA 1) , volume 3 of
Documents Math´ematiques (Paris) [Mathematical Documents(Paris)] . Soci´et´e Math´ematique de France, Paris, 2003. S´eminaire de g´eom´etrie alg´ebrique du Bois Marie 1960–61. [AlgebraicGeometry Seminar of Bois Marie 1960-61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated andannotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 ∞ -categories. arXiv preprint , 2010. arXiv:1007.2925v2. (On p. 1)[GS07] Paul Goerss and Kristen Schemmerhorn. Model categories and simplicial methods. In Interactions between homotopy theoryand algebra , volume 436 of
Contemp. Math. , pages 3–49. Amer. Math. Soc., Providence, RI, 2007. (On p. 9)[Hir03] Philip S. Hirschhorn.
Model categories and their localizations , volume 99 of
Mathematical Surveys and Monographs . AmericanMathematical Society, Providence, RI, 2003. (On pp. 8 and 9)[HKRS17] Kathryn Hess, Magdalena Ke¸dziorek, Emily Riehl, and Brooke Shipley. A necessary and sufficient condition for inducedmodel structures.
J. Topol. , 10(2):324–369, 2017. (On p. 25)[HM15] Gijs Heuts and Ieke Moerdijk. Left fibrations and homotopy colimits.
Math. Z. , 279(3-4):723–744, 2015. (On pp. 3 and 4) [HM16] Gijs Heuts and Ieke Moerdijk. Left fibrations and homotopy colimits ii. arXiv preprint , 2016. arXiv:1602.01274v1. (On p. 3)[Joh02] Peter T. Johnstone.
Sketches of an elephant: a topos theory compendium. Vol. 1 , volume 43 of
Oxford Logic Guides . The ClarendonPress, Oxford University Press, New York, 2002. (On pp. 1 and 16)[Joy08a] Andr´e Joyal. Notes on quasi-categories. preprint , 2008. Unpublished notes (accessed 08.02.2021). (On pp. 2, 3, and 10)[Joy08b] Andr´e Joyal. The theory of quasi-categories and its applications. 2008. Unpublished notes (accessed 08.02.2021). (On pp. 2,3, and 10)[JT07] Andr´e Joyal and Myles Tierney. Quasi-categories vs Segal spaces. In
Categories in algebra, geometry and mathematical physics ,volume 431 of
Contemp. Math. , pages 277–326. Amer. Math. Soc., Providence, RI, 2007. (On pp. 3, 6, 12, and 13)[Lur04] Jacob Lurie.
Derived algebraic geometry . ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–Massachusetts Institute of Tech-nology. (On p. 1)[Lur09a] Jacob Lurie.
Higher topos theory , volume 170 of
Annals of Mathematics Studies . Princeton University Press, Princeton, NJ,2009. (On pp. 1, 2, 3, 4, 5, 9, 10, 14, 15, 16, 17, 21, 22, 26, 30, and 31)[Lur09b] Jacob Lurie. (infinity, 2)-categories and the goodwillie calculus i. arXiv preprint , 2009. arXiv:0905.0462v2. (On p. 5)[Lur09c] Jacob Lurie. On the classification of topological field theories. In
Current developments in mathematics, 2008 , pages 129–280.Int. Press, Somerville, MA, 2009. (On p. 5)[Lur17] Jacob Lurie. Higher algebra. Unpublished notes (accessed 08.02.2021), September 2017. (On p. 10)[nla20] model structure for cartesian fibrations. nLab, 2020. URL:nlab (version: 08.02.2021). (On p. 16)[Qui67] Daniel G. Quillen.
Homotopical algebra . Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin-New York, 1967. (Onpp. 2, 7, and 9)[Ras17a] Nima Rasekh. Cartesian fibrations and representability. arXiv preprint , 2017. arXiv:1711.03670v3. (On p. 6)[Ras17b] Nima Rasekh. Yoneda lemma for simplicial spaces. arXiv preprint , 2017. arXiv:1711.03160v3. (On pp. 3, 13, 14, and 27)[Ras21] Nima Rasekh. Cartesian fibrations of complete Segal spaces. arXiv preprint , 2021. arXiv:2102.05190v2. (On pp. 4, 5, 6, 14,and 15)[Ree74] Christopher Leonard Reedy.
HOMOLOGY OF ALGEBRAIC THEORIES . ProQuest LLC, Ann Arbor, MI, 1974. Thesis(Ph.D.)–University of California, San Diego. (On p. 8)[Rez01] Charles Rezk. A model for the homotopy theory of homotopy theory.
Trans. Amer. Math. Soc. , 353(3):973–1007, 2001. (Onpp. 3, 9, and 11)[Rez10] Charles Rezk. A Cartesian presentation of weak n -categories. Geom. Topol. , 14(1):521–571, 2010. (On pp. 3 and 5)[RV17] Emily Riehl and Dominic Verity. Fibrations and Yoneda’s lemma in an ∞ -cosmos. J. Pure Appl. Algebra , 221(3):499–564,2017. (On pp. 2, 3, 12, and 13)[RV21] Emily Riehl and Dominic Verity. Elements of ∞ -category theory. 2021. Unpublished book (01.02.2021). (On p. 31)[sga72] Th´eorie des topos et cohomologie ´etale des sch´emas. Tome 1: Th´eorie des topos . Lecture Notes in Mathematics, Vol. 269. Springer-Verlag, Berlin-New York, 1972. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1963–1964 (SGA 4), Dirig´e par M. Artin,A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. (On p. 1)[Ste17] Danny Stevenson. Covariant model structures and simplicial localization.
North-West. Eur. J. Math. , 3:141–203, 2017. (Onp. 3)[To¨e05] Bertrand To¨en. Vers une axiomatisation de la th´eorie des cat´egories sup´erieures.
K-Theory , 34(3):233–263, 2005. (On p. 3)[Ver08] D. R. B. Verity. Weak complicial sets. I. Basic homotopy theory.
Adv. Math. , 219(4):1081–1149, 2008. (On p. 5)´E
COLE P OLYTECHNIQUE
F ´ ED ´ ERALE DE L AUSANNE , SV BMI UPHESS, S
TATION
8, CH-1015 L
AUSANNE , S
WITZERLAND
Email address ::