Spahn HTT, A.3 simplicial categories (Rev #7)

This is a subentry of a reading guide to HTT.

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Remark

(morphisms in enriched categories)

In a model category AA there are stricly speaking no morphism defined but only hom objects. So if we wish to define the notion of an enriched model category where we expect to have distinguished classes of morphisms we need to refer to an associated (ordinary - i.e. SetSet-enriched) model category where we have morphisms and to qualify our morphisms there as cofibrations, fibrations and weak equivalences, respectively. This is explicated in the following way:

Definition

(Street, Chapter 1.3)

Let VV be a monoidal category. Let V 0V_0 denote the set of objects of VV. Let \mathcal{I} denote the terminal VV-category :={0,I}\mathcal{I}:=\{0,I\}; i.e. \mathcal{I} has precisely one object 00 and the monoidal unit is defined to be the hom object I:=hom(0,0)I:=hom(0,0). Let ** denote the terminal category. Let V:=V 0(I,):V 0SetV:=V_0(I,-):V_0\to Set. Let VCatV Cat denote the 2-category of VV-categories. Then there is a functor

() 0:=VCat(,):{VCatCat AidA (f:IA(a,b))(*VA(a,b))(-)_0:=V Cat(\mathcal{I},-):\begin{cases} V Cat\to Cat \\ A\stackrel{id}{\mapsto}A \\ (f:I\to A(a,b))\mapsto (*\to V A(a,b)) \end{cases}

called the underlying set functor.

So if we speak of a cofibration, fibration or weak equivalences f:abf:a\to b in an enriched category AA we mean in fact () 0(f:IA(a,b))(-)_0(f:I\to A(a,b)).

A.3.1 Enriched monoidal model categories

Definition A.3.1.1

(Quillen bifunctor)

Let A,B,CA,B,C be model categories.

A functor F:A×BCF:A\times B\to C is called Quillen bifunctor if the following conditions are satisfied:

(1) For cofibrations i:aa i:a\to a^\prime, and j:bb j:b\to b^\prime in AA resp. in BB, the induced map

ij:F(a ,b) F(a,b)F(a,b )F(a ,b )i\wedge j:F(a^\prime, b) \coprod_{F(a,b)}F(a,b^\prime)\to F(a^\prime,b^\prime)

is a cofibration in CC. Moreover iji\wedge j is acyclic if either ii or jj is acyclic; where the pushout is

F(a,b) F(Id,j) F(a,b ) F(i,Id) F(a ,b) F(a ,b) F(a,b)F(a,b )\array{ F(a,b) &\stackrel{F(Id,j)}{\to}& F(a,b^\prime) \\ \;\;\downarrow^{F(i,Id)} && \downarrow \\ F(a^\prime,b) &\stackrel{}{\to}& F(a^\prime,b) \coprod_{F(a,b)} F(a,b^\prime) }

(2) FF preserves small colimits in each variable seperately.

Remark

setting i:0c i:0\hookrightarrow c^\prime shows that condition 1. in the previous definition reduces to the requirement on F(c ,)F(c^\prime,-) to preserve cofibrations and acyclic cofibrations.

Definition A.3.1.2

(monoidal model category)

A monoidal model category is a monoidal category SS equipped with a model structure satisfying the following:

  1. The tensor product :S×SS\otimes:S\times S\to S is a left Quillen bifunctor.

  2. The unit object 1S1\in S is cofibrant.

  3. The monoidal structure is closed.

Example A.3.1.4

The category sSetsSet is a monoidal model category with respect to the cartesian product and the Kan model structure.

Definition A.3.1.5

(SS-enriched model category)

Let SS be a monoidal model category.

A SS-enriched model category is defined to be a SS-enriched category AA equiped with a model structure satisfying the following:

  1. AA is tonsured and cotensored over SS.

  2. The tensor product :A×SA\otimes:A\times S\to A is a left Quillen bifunctor

Remark A.3.1.6

(alternative characterization of the Quillen bifunctor :A×SA\otimes:A\times S\to A)

Proposition A.3.1.10

Let CC, DD be SS-enriched model categories. Let (FG):DGD(F\dashv G):D\stackrel{G}{\to} D be a Quillen adjunction between the underlying model categories. Let every object of CC be cofibrant. Let

β x,s:sF(x)F(sx)\beta_{x,s}: s\otimes F(x)\to F(s\otimes x)

be a weak equivalence for every pair of cofibrant objects xCx\in C, sSs\in S. Then the following are equivalent:

  1. (FG)(F\dashv G) is a Quillen equivalence.

  2. The restriction of GG determines a weak equivalence of SS-enriched categories D C D^\circ\to C^\circ.

Corollary A.3.1.12

Let (FG):DGD(F\dashv G):D\stackrel{G}{\to} D be a Quillen equivalence between simplicial model categories where every object of CC is cofibrant. Let GG be a simplicial functor. Then GG induces an equivalence of \infty-categories N(D )N(C )N(D^\circ)\to N(C^\circ).

A.3.2 The model structure on SS-enriched categories

Definition A.3.2.1

Let SS be a monoidal model category.

A functor F:CC F:C\to C^\prime in sSetCatsSet Cat is a weak equivaleence if the induced functor hChC hC\to h C^\prime is an equivalence of hSh S-enriched categories.

In other words: F is a weak equivalence iff

(1) For every pair X,YCX,Y\in C, the induced map

Map C(X,Y)Map C (F(X),F(Y))Map_C (X,Y)\to Map_{C^\prime} (F(X), F(Y))

is a weak equivalence in SS.

(2) FF is essentially surjective on the level of homotopy categories.

The following definition says a functor between categories is called a quasi fibrations if every isomorphism has a lift with respect to FF.

Definition A.3.2.7

Let FF:C\to D$ be a functor between classical categories.

FF is called a quasi-fibration if, for every object xCx\in C and every isomorphism f:F(x)yf:F(x)\to y in DD, there exists an isomorphism f¯:xy¯\overline f:x\to \overline y in CC such that F(f)=fF(f)=f.

Theorem 3.2.24

Let SS be an excellent model category. Then:

  1. An SS-enriched category CC is a fibrant object of sSetCatsSet Cat iff it is locally fibrant: i.e. for all X,YCX,Y\in C the hom object Map C(X,Y)SMap_C (X,Y)\in S is fibrant.

  2. Let F:CDF:C\to D be a SS-enriched functor where DD is a fibrant object of sSetCatsSet Cat. Then FF is a fibration iff FF is a local fibration.

Definition

Let SS be a monoidal category. Let CC be an SS-enriched category.

(1) A morphism ff in CC is called an equivalence if the homotopy class [f][f] of ff is an isomorphism in hCh C.

(2) CC is called locally fibrant object if for every pair of objects X,YCX,Y\in C, the mapping space Map C(X,Y)Map_C(X,Y) is a fibrant object of SS.

(3) An SS-enriched functor F:CC F:C\to C^\prime is called a local fibration if the following conditions are satisfied:

(3.i) Map C(X,Y)Map C (FX,FY)Map_C (X,Y)\to Map_{C^\prime} (FX,FY) is a fibration in SS for every X,YCX,Y\in C.

(3.ii) The induced map hChC h C\to h C^\prime is a quasi-fibration of categories.

Definition A.3.2.16

(excellent model category)

A model category SS is called excellent model category if it is equipped with a symmetric monoidal structure and satisfies the following conditions

(A1) SS is combinatorial.

(A2) Every monomorphism in SS is a cofibration and the collection of cofibrations in SS is stable under products.

(A3) The collection of weak equivalencies in SS is stable under filtered colimits.

(A4) :S×SS\otimes:S\times S\to S is a Quillen bifunctor.

(A5) The monoidal model category SS satisfies the invertibility hypothesis.

A.3.3 Model structures on diagram categories

A.3.4 Path spaces in SS-enriched categories

A.3.5 Homotopy colimits in SS-enriched categories

A.3.6 Exponentiation in model categories

A.3.7 Localizations of simplicial model categories

References

Revision on June 27, 2012 at 00:02:51 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.