Spahn
HTT, A.3 simplicial categories

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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

HTT, A.3.1

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

HTT, A.3.2

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

Last revised on June 29, 2012 at 22:26:09. See the history of this page for a list of all contributions to it.