Spahn HTT, A.3.1 (changes)

Showing changes from revision #1 to #2: Added | Removed | Changed

This is a subentry of HTT, A.3 and of a reading guide to HTT.

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.

The following remark explicates the relation between simplicial homotopy theory and model-category-theoretic homotopy theory.

Corollary Remark A.3.1.12 A.3.1.8

Let (CFG):DGD (F\dashv C G):D\stackrel{G}{\to} D be a Quillen equivalence between simplicial model categories category, where let every object of C X C X is be cofibrant. a Let cofibrant object of G C G C , be let a simplicial functor. Then G Y G Y induces be an a equivalence fibrant object ofC \infty C -categories . Then we haveN(D )N(C )N(D^\circ)\to N(C^\circ).

(1) Map C(X,Y)Map_C (X,Y) is a Kan complex.

(2) hom hC(X,Y)π 0Map C(X,Y)hom_hC(X,Y)\simeq \pi_0 Map_C (X,Y)

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).

Last revised on June 29, 2012 at 23:37:20. See the history of this page for a list of all contributions to it.