Spahn HTT, A.3 simplicial categories (Rev #10, changes)

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This is a subentry of a reading guide to HTT.

Contents

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 ℐ\mathcal{I} denote the terminal VV-category ℐ:={0,I}\mathcal{I}:=\{0,I\}; i.e. ℐ\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 0→SetV:=V_0(I,-):V_0\to Set. Let VCatV Cat denote the 2-category of VV-categories. Then there is a functor

(−) 0:=VCat(ℐ,−):{VCat→Cat A↦idA (f:I→A(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:a→bf:a\to b in an enriched category AA we mean in fact (−) 0(f:I→A(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×B→CF:A\times B\to C is called Quillen bifunctor if the following conditions are satisfied:

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

i∧j: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 i∧ji\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.

HTT, A.3.1
Remark Definition A.3.1.1

setting (Quillen bifunctor)i:0↪c ′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.

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

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

(1) For cofibrations i:a→a ′i:a\to a^\prime, and j:b→b ′j:b\to b^\prime in AA resp. in BB, the induced map
i∧j: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 i∧ji\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.
Definition Remark A.3.1.2

(monoidal setting model category)i:0↪c ′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.

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

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

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

  3. The monoidal structure is closed.
Example Definition A.3.1.4 A.3.1.2

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

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

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

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

  3. The monoidal structure is closed.
Definition Example A.3.1.5 A.3.1.4

( The category S sSet S sSet -enriched is a monoidal model category) category with respect to the cartesian product and the Kan model structure.

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×S→A\otimes:A\times S\to A is a left Quillen bifunctor
Remark Definition A.3.1.6 A.3.1.5

(alternative ( characterization of the Quillen bifunctor⊗:A×S→A \otimes:A\times S S\to A ) -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×S→A\otimes:A\times S\to A is a left Quillen bifunctor
Proposition Remark A.3.1.10 A.3.1.6

Let (alternative characterization of the Quillen bifunctorC⊗:A×S→A C \otimes:A\times S\to A , )DD be SS-enriched model categories. Let (F⊣G):D→GD(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:s⊗F(x)→F(s⊗x)\beta_{x,s}: s\otimes F(x)\to F(s\otimes x)

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

  1. (F⊣G)(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 Proposition A.3.1.12 A.3.1.10

Let (CF⊣G):D→GD (F\dashv C G):D\stackrel{G}{\to} D , be a Quillen equivalence between simplicial model categories where every object of C D C D is be cofibrant. Let G S G S -enriched be model a categories. simplicial Let functor. Then(F⊣G):D→GD G (F\dashv G):D\stackrel{G}{\to} D induces be an a equivalence Quillen adjunction between the underlying model categories. Let every object of∞C \infty C -categories be cofibrant. LetN(D ∘)→N(C ∘)N(D^\circ)\to N(C^\circ).

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

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

  1. (F⊣G)(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.

A.3.2 The model structure on SS-enriched categories

Corollary A.3.1.12

Let (F⊣G):D→GD(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

Revision on June 29, 2012 at 22:26:09 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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