Spahn
HTT, A.2.2

Definition

Let XX be an object in a model category.

  1. A cylinder object is defined to be a factorization of the codiagonal map XXXX\coprod X\to X for XX into a cofibration followed by a weak equivalence.

  2. A path object is defined to be a factorization of the diagonal map XX×XX\to X\times X for XX into a weak equivalence followed by a fibration .

Proposition A.2.2.1

Let CC be a model category. Let XX be a cofibrant object of CC. Let YY be a fibrant object of CC. Let f,g:XYf,g:X\to Y be two parallel morphisms. Then the following conditions are equivalent.

  1. The coproduct map fgf\coprod g factors through every cylinder object for XX.

  2. The coproduct map fgf\coprod g factors through some cylinder object for XX.

  3. The product map f×gf\times g factors through every path object for YY.

  4. The product map f×gf\times g factors through some path object for YY.

Definition

(homotopy, homotopy category of a model category)

Let CC be a model category.

(1) Two maps f,g:XYf,g:X\to Y from a cofibrant object to a fibrant object satisfying the conditions of Proposition A.2.2.1 are called homotopic morphisms. Homotopy is an equivalence relation \simeq on hom C(X,Y)hom_C (X,Y).

(2) The homotopy category hCh C of CC is defined to have as objects the fibrant-cofibrant objects of CC. The hom objects hom hC(X,Y)hom_{hC}(X,Y) are defined to be the set of \simeq equivalence classes of hom C(X,Y)hom_C (X,Y).

The following remark gives an alternative equivalent definition of the homotopy category of a model category:

Remark and Definition

The homotopy category hChC (more precisely the projection map Q:ChCQ:C\to hC) is couniversal in the following sense:

  • for any (possibly large) category AA and functor F:CAF : C \to A such that FF sends all wWw \in W to isomorphisms in AA, there exists a functor F Q:hCAF_Q : hC \to A and a natural isomorphism
C F A Q F Q hC \array{ C &&\stackrel{F}{\to}& A \\ \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q} \\ hC }

The second condition implies that the functor F QF_Q in the first condition is unique up to unique isomorphism.

As always is the the case with (co)universal properties the object in question can be defined by this property.

Last revised on June 29, 2012 at 22:15:07. See the history of this page for a list of all contributions to it.