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 $A$ 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

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

HTT, A.3.1

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:

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

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

3. 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:

1. $A$ is tonsured and cotensored over $S$.

2. 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:

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

2. 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 $N(D^\circ)\to N(C^\circ)$.

HTT, A.3.2

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.