deformation retract of a homotopical category



Deformations and deformation retracts are tools in homotopy theory for constructing the homotopy category of a model category or more general homotopical category.

The idea of a deformation retract is to find a full subcategory of a given homotopical category CC such that

  • a given functor FF from CC to another homotopical category becomes a homotopical functor on this subcategory;

  • every object in the category is naturally weakly equivalent to an object in the subcategory.

Deformations are a generalizations of cofibrant replacement functors in a model category.


Let CC be a homotopical category.

  • A left deformation of CC is a functor Q:CCQ : C \to C equipped with a natural weak equivalence

    Q C q C Id \array{ & \nearrow \searrow^{Q}& \\ C &\Downarrow^{q}_\simeq& C \\ & \searrow \nearrow_{Id} }

    (it follows from 2-out-of-3 that QQ is a homotopical functor).

  • A left deformation retract is a full subcategory C QC_Q containing the image of a left deformation (Q,q)(Q,q).

Now let F:CDF : C \to D be a functor between homotopical categories.

  • A left deformation retract for FF is a left deformation retract C QC_Q of CC such that FF becomes a homotopical functor when restricted to C QC_Q.

Right deformations are defined analogously.


There are pretty obvious generalizations of deformation retracts for functors of more than one variable.

  • A deformation retract for a two-variable adjunction (,hom l,hom r):C×DE(\otimes , hom_l, hom_r) : C \times D \to E consists of left deformation retracts C QC_Q, D QD_Q for CC and DD, respectively, and a right deformation retract E QE_Q of EE, such that

    • \otimes is homotopical on C Q×D QC_Q \times D_Q;

    • hom lhom_l is homotopical on C Q op×E QC_Q^{op} \times E_Q;

    • hom rhom_r is homotopical on D Q op×E QD_Q^{op} \times E_Q.


The definition of deformation and deformation retract is in paragraph 40 of

  • William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith. Homotopy Limit Functors on Model Categories and Homotopical Categories, volume 113 of Mathematical Surveys and Monographs. American Mathematical Society, 2004.

The notion of deformation retract of a two-variable adjunction is definition 15.1, p. 43 in

  • Michael Shulman, Homotopy limits and colimits and enriched homotopy theory (arXiv)

Last revised on November 17, 2009 at 18:35:29. See the history of this page for a list of all contributions to it.