Spahn HTT, A.3 simplicial categories (Rev #8, changes)

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This is a subentry of a reading guide to HTT.

A.3.1 Enriched monoidal model categories
A.3.2 The model structure on $S$ -enriched categories
A.3.3 Model structures on diagram categories
A.3.4 Path spaces in $S$ -enriched categories
A.3.5 Homotopy colimits in $S$ -enriched categories
A.3.6 Exponentiation in model categories
A.3.7 Localizations of simplicial model categories
References

Remark

(morphisms in enriched categories)

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

(-)_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 $f:a\to b$ in an enriched category $A$ we mean in fact $(-)_0(f:I\to A(a,b))$ .

A.3.1 Enriched monoidal model categories

Definition A.3.1.1

(Quillen bifunctor)

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

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

(1) For cofibrations $i:a\to a^\prime$ , and $j:b\to b^\prime$ in $A$ resp. in $B$ , the induced map

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 $C$ . Moreover $i\wedge j$ is acyclic if either $i$ or $j$ is acyclic; where the pushout is

\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) $F$ preserves small colimits in each variable seperately.

Remark

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

Definition A.3.1.2

(monoidal model category)

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

The tensor product $\otimes:S\times S\to S$ is a left Quillen bifunctor.
The unit object $1\in S$ is cofibrant.
The monoidal structure is closed.

Example A.3.1.4

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

Definition A.3.1.5

( $S$ -enriched model category)

Let $S$ be a monoidal model category.

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

$A$ is tonsured and cotensored over $S$ .
The tensor product $\otimes:A\times S\to A$ is a left Quillen bifunctor

Remark A.3.1.6

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

Proposition A.3.1.10

Let $C$ , $D$ be $S$ -enriched model categories. Let $(F\dashv G):D\stackrel{G}{\to} D$ be a Quillen adjunction between the underlying model categories. Let every object of $C$ be cofibrant. Let

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

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

$(F\dashv G)$ is a Quillen equivalence.
The restriction of $G$ determines a weak equivalence of $S$ -enriched categories $D^\circ\to C^\circ$ .

Corollary A.3.1.12

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

A.3.2 The model structure on $S$ -enriched categories

Definition A.3.2.1

Let $S$ be a monoidal model category.

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

In other words: F is a weak equivalence iff

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

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

is a weak equivalence in $S$ .

(2) $F$ 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 $F$ .

Definition A.3.2.7

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

$F$ is called a quasi-fibration if, for every object $x\in C$ and every isomorphism $f:F(x)\to y$ in $D$ , there exists an isomorphism $\overline f:x\to \overline y$ in $C$ such that $F(f)=f$ .

Theorem 3.2.24

Let $S$ be an excellent model category. Then:

An $S$ -enriched category $C$ is a fibrant object of $sSet Cat$ iff it is locally fibrant: i.e. for all $X,Y\in C$ the hom object $Map_C (X,Y)\in S$ is fibrant.
Let $F:C\to D$ be a $S$ -enriched functor where $D$ is a fibrant object of $sSet Cat$ . Then $F$ is a fibration iff $F$ is a local fibration.

Definition

Let $S$ be a monoidal category. Let $C$ be an $S$ -enriched category.

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

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

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

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

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

Definition A.3.2.16

(excellent model category)

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

(A1) $S$ is combinatorial.

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

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

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

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

A.3.3 Model structures on diagram categories

A.3.4 Path spaces in $S$ -enriched categories

A.3.5 Homotopy colimits in $S$ -enriched categories

A.3.6 Exponentiation in model categories

A.3.7 Localizations of simplicial model categories

References

Ross Street, basic concepts of enriched category theory, pdf
enriched model category

Revision on June 27, 2012 at 13:33:29 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn HTT, A.3 simplicial categories (Rev #8, changes)

Contents

Remark

Definition

A.3.1 Enriched monoidal model categories

Definition A.3.1.1

Remark

Definition A.3.1.2

Example A.3.1.4

Definition A.3.1.5

Remark A.3.1.6

Proposition A.3.1.10

Corollary A.3.1.12

A.3.2 The model structure on SS-enriched categories

Definition A.3.2.1

Definition A.3.2.7

Theorem 3.2.24

Definition

Definition A.3.2.16

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

A.3.2 The model structure on $S$ -enriched categories

A.3.4 Path spaces in $S$ -enriched categories

A.3.5 Homotopy colimits in $S$ -enriched categories