Comparison to Other Approaches

The aim of this section is to compare the quasi-categorical approach to the theory of com-plete Segal spaces. In this thesis, this will only be used to transfer results by Julie Bergner about homotopy limits of complete Segal spaces to homotopy limits of quasi-categories, so it might be skipped in first reading.

Before explaining the theory of complete Segal spaces, we introduce the theory of relative categories. Arelative categoryis a categoryC equipped with a chosen subcategory W (called the sub category ofweak equivalences) which contains all objects ofC. Important examples are model categories with their weak equivalences.

1More precisely,X∈sSet is an∞-category iff Map(∆²,X)→Map(Λ²₁,X)is an acyclic Kan fibration.

59 Given a relative category C, we can construct a simplicial category L^HC^{, thehammock} localization(for a definition, see [DK80b, Section 3.1]). We get a diagram

ModCat ^//relCat

L^H

sModCat ⁽⁾

◦ //

OO

sCat

This commutes up to a natural zig zag of Dwyer–Kan equivalences of simplicial categories by [DK80b, Proposition 4.8].

Definition 4.1.3. A relative functor between relative categories (C^,W) and (C^0,W⁰) is a functor F: C → C^{0 such that F}(W)⊂ W^{0. It is a}homotopy equivalenceif there is a relative functor G: C⁰ → C ^{such that FGand GF}are naturally equivalent to the identity functors (i.e., there is a zig zag of natural transformations consisting of weak equivalences).

Lemma 4.1.4. A homotopy equivalence F: (C^,W)→(C^0,W⁰)induces a Dwyer-Kan equivalence L^HF: L^H(C^,W)→ ^LH(C^0,W⁰).

Proof. By [DK80b, Proposition 3.2], the homotopy category Ho(L^HC)is a localization ofC at the class of weak equivalences. Thus, Ho(L^HF): Ho(L^HC) → ^Ho(L^HC)is essentially surjective.

Therefore, it is enough to show the following: Suppose that I and J are relative endo-functors of(C^,W)with a natural transformations: I → ^J consisting of weak equivalences between them. Then L^HC(X,Y) → ^LHC(IX,IY) is a weak equivalence iff L^HC(X,Y) → L^HC(JX,JY)is a weak equivalence.

By [DK80a, Proposition 3.5], we have a commutative diagram L^HC(IX,IY)

s_∗

((P

PP PP PP PP PP P L^HC(X,Y)

77o

oo oo oo oo oo

''O

OO OO OO OO

OO L^HC(IX,JY)

L^HC(JX,JY)

sn^∗nnnnnn66 nn

nn n

By [DK80a, Proposition 3.3], s_∗ ands^∗are weak equivalences. Thus, the result.

We equip the category of relative categories (with relative functors between them) with the model structure from [BK12b], which we call the Barwick–Kan model structure, and denote it by RelCat. With the weak equivalences of the Bergner respectively Barwick–Kan model structures, both the category of simplicial categories and the category of relative categories get the structures of relative categories.

Proposition 4.1.5([BK12a], Theorem 1.7). The Hammock localization is a homotopy equivalence between the relative categories of relative categories and simplicial categories.

The theory of Segal spaces begins with the observation that for the nerve NC ^{of a} categoryC, we have an isomorphism

(NC)n→(NC)1×(NC)₀ · · · ×(NC)₀ (NC)1

whose inverse is given by composition.

Definition 4.1.6. A simplicial space² W is called a Segal space if it is Reedy fibrant and

2Here, ‘space‘ stands for a ‘simplicial set‘.

the Segal mapW_n →^W1×W0 · · · ×W0W₁ is a weak equivalence of simplicial sets. A Segal space is said to becompleteif the Rezk completion map is an equivalence (see [Rez01, §4-6]

for details).

For two 0-simplices x,y ∈^W0 in a (Reedy fibrant) simplicial spaceW, define the map-ping space map_W(x,y)to be the fiber of the map(d₁,d0): W₁→^W0×^W0 over(x,y). The homotopy category Ho(W) hasW_0,0 as objects andπ₀mapW(x,y)as Hom-sets. We say that a map of (Reedy fibrant) simplicial spaces is aDwyer–Kan equivalenceif it induces an equivalence of homotopy categories and weak equivalences of mapping spaces.

The category of simplicial spaces can be equipped with a (simplicial) model structure ([Rez01, Theorem 7.2]) such that the fibrant objects are exactly the complete Segal spaces, the weak equivalences between Segal spaces are given by Dwyer–Kan equivalences and every object is cofibrant. This model structure is Quillen equivalent both to the Joyal and the Bergner model structure. For example, we have a Quillen equivalence:

sSet

p^∗₁ //

ssSet

i^∗₁

oo

Here, the two Quillen functors a induced by the projection p₁: ∆×^∆ → ^∆ to the first coordinate and the mapi₁: ∆→∆×∆sending[n]to([n],[0]).

We can associate to every relative category a simplicial space as follows: Let C^{[n] be} category of chains of n composable morphisms inC. A morphism between two chains is called a weak equivalence if it is a weak equivalence on every object. Then, we define a simplicial space N(C^,W)by

N(C^,W)n= N(we(C^[n])), which is called theclassifying diagram.

Theorem 4.1.7([BK12b], Theorem 6.1 and Key Lemma 5.4). There is a Quillen equivalence ssSet

K_ξ //

RelCat

Nξ

oo

such that there is a natural transformation N → ^Nξ which consists of Reedy equivalences (which are, in particular, equivalences in the Rezk model structure). In addition, N_ξ(f)is a weak equiva-lence (fibration) iff f is a weak equivaequiva-lence (fibration), for f a morphism inssSet. In particular, the right derived functor RN_ξ is weakly equivalent to N.

Next, we want to discuss an amazing result by Toën.

Theorem 4.1.8 ([Toë05],Theorem 6.3). For C a simplicial category, denote by RAut(C) the simplicial monoid consisting of those components of the derived mapping space Map_sCat(C^,C) consisting of Dwyer–Kan equivalences. Then there is a weak equivalence of simplicial monoids RAut(L^HssSet) ' ^C2. An endomorphism F of L^HssSetlies in the component of the identity iff there is weak equivalence between the diagrams

F(∆⁰_δ) ^//// F(∆¹_δ) and ∆⁰_δ ^////∆¹_δ .

Here,∆⁰_δ and∆¹_δ are∆⁰and∆¹viewed as discrete simplicial spaces and the maps are (induced by) the inclusion of the end points.

61 Note that if

C ^{F //}D

oo G

is a Quillen equivalence, the derived functors LF andRG define a homotopy equivalence of relative categories and hence Dwyer–Kan equivalences of the Hammock localizations L^HC ^andLHD by Lemma 4.1.4.

An object X∈ssSet can be viewed as an object inL^HssSet. By [DK80b, 4.8], we have a zig zag of Dwyer–Kan equivalences

L^HssSet ^//diagL^HssSet^oo ssSet^◦

which are all identity on objects. We choose a fibrant replacement functor()^f in simplicial categories, which preserves objects. One possibility is to apply to apply S_•|| ^{to every} morphism space (where S_• denotes the singular complex). This is functorial an we get a zig zag

(L^HssSet)^{f //}(diagL^HssSet)^{f oo} ssSet^◦ Applying nerves, we get a zig zag of equivalences

N((L^HssSet)^f) ^//N((diagL^HssSet)^f)

G_⁰_//

_ N(ssSet^◦)

G

oo

using the Ken Brown lemma ([Hov99, 1.1.12]). The dashed arrow is an inverse weak equivalence toG, which exists since both nerves are bifibrant. The map Qfrom the objects of ssSet to the 0-simplices ofN((diagL^HssSet)^f)is the identity. We denote the composition FQ byκ. Given X ∈ ^{ssSet◦, we have QX} = GX. Since FG is equivalent to the identity functor,κX is naturally equivalent toX. Note also that κ preserves equivalences between objects (here, an equivalence is a morphism inducing an isomorphism in the homotopy category).

Corollary 4.1.9. With this notation, there is a natural equivalence between κ(p^∗₁N(M^◦)) and κ(N(M^,W))in N(ssSet^◦).

Proof. By Proposition 4.1.5 and the fact that p^∗₁ and N are Quillen equivalences, F = Lp^∗₁RNL^H defines a homotopy equivalence between the relative categories RelCat and ssSet (with the Rezk model structure). By Theorem 4.1.7, N(−^,−) is also a homotopy equivalence from RelCat to ssSet. Thus, N(−^,−)K_ξ and FK_ξ are auto homotopy equiva-lences of ssSet (note that every object of ssSet is cofibrant). By Lemma 4.1.4, they define auto Dwyer–Kan equivalences of L^HssSet. By [BK12b, Proposition 7.3],K_ξ([n]) ' ^K([n]), where K([n])is the relative category [n]where weak equivalences are just identities. The Hammock localization ofK([n])is just the discrete category[n]andp^∗₁N([n]) = p^∗₁∆ⁿ =∆ⁿ_δ in ssSet. Similarly, we get that N([n], id_[n]) =∆ⁿ_δ. Both identifications are compatible with the structure maps in the category∆. Thus,FK_ξ andN(−^,−)K_ξ lie in the same path com-ponents in the derived mapping space Map_sCat(L^HssSet,L^HssSet). SinceK_ξ is a Dwyer–

Kan equivalence, alsoFandN(−^,−)lie in the same path component of the derived map-ping space Map_sCat(L^HRelCat,L^HssSet). If we postcompose with the fibrant replacement

L^HssSet→(L^HssSet)^f, bothFand N(−^,−)factor over the fibrant replacement:

L^HRelCat

F //

N(−,−)//(L^HssSet)^f

(L^HRelCat)^f

F⁰

==z

zz zz zz zz

N⁰(−^,−)

==z

zz zz zz zz

We denote these maps (L^HRelCat)^f → (L^HssSet)^f by F⁰ and N⁰(−^,−). The induced maps NF⁰ and NN⁰(−^,−) from N((L^HRelCat)^f)to N((L^HssSet)^f)lie in the same path component of

Map_sSet(N((L^HRelCat)^f),N((L^HssSet)^f))

(which is, at the same time, the derived mapping space since all objects in sSet are cofi-brant and N((L^HssSet)^f) is fibrant). Thus, also NF and NN(−^,−) lie in the same path component of Map_sSet(N(L^HRelCat),N((L^HssSet)^f)))andκNFandκNN(−^,−)lie in the same path component of

Map_sSet(N(L^HRelCat),N(ssSet^◦)),

i.e., there is a natural equivalence betweenF(M^,W)andN(M^,W)in NssSet^◦.

The simplicial categoryM^◦ is a fibrant replacement ofL^HMand all objects of sSet are cofibrant. ThusF(M^,W) = Lp₁^∗RNL^H(M) = p^∗₁N(M^◦)and the result follows.

LetsModCatbe the category of simplicial model categories where morphisms are given by simplicial functors preserving fibrations, cofibrations and weak equivalences. Further-more, we denote by holim the derived functor of the limit in a model category. The following corollary owes much to Chris Schommer-Pries.

Corollary 4.1.10. Let I →^sModCat,i7→ Mi be a diagram of simplicial model categories. Then p^∗₁holimIN(M^◦_i)is weakly equivalent toholimIN(Mi,Wi); here, the homotopy limits are built in the Joyal model structure onsSetand the Rezk model structure onssSetrespectively. In partic-ular, for a simplicial model categoryM, the nerve NM^◦ is weakly equivalent toholim_IN(M^◦_i) iff N(M^,W)is weakly equivalent toholimIN(Mi,Wi).

Proof. We have two diagramsκ(p^∗₁NM^◦_i)andκ(N(Mi,Wi))of the formN I→ ^N(ssSet^◦), which are homotopic by the last corollary. These are naturally equivalent to (p^∗₁NM^◦_i)^f and (N(Mi,Wi))^f, where ()^f denotes fibrant replacement in the Rezk model structure.

Thus, holimN I(p^∗₁NM^◦_i)^f ' ^holimN I(N(Mi,Wi))^f, where the homotopy limit is taken in the∞-categorical sense. By [Lur09b, Theorem 4.2.4.1], holimI(p₁^∗NM^◦_i)'^holimI(p^∗₁NM^◦_i)^f (in the model categorical sense), where the diagram is in ssSet, is equivalent to holim_{N I}(p^∗₁NM^◦i)^f and holimIN(Mi,Wi)to holimN I(N(Mi,Wi))^f as well. Since p^∗₁ is (the derived functor of) a Quillen equivalence, p^∗₁holim_INM^◦_i is weakly equivalent to holim_Ip^∗₁NM^◦_i ^{and the} first statement follows.

For the second, note that NM^◦ '^holimIN(M^◦_i)iff

N(M^,W)' ^p∗1NM^◦' ^p∗1holim_IN(M^◦i)'^holimIN(Mi,Wi).

63

Im Dokument United Elliptic Homology (Seite 58-63)