nLab deformation retract of a homotopical category

Idea

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 $C$ such that

• a given functor $F$ from $C$ 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.

Definition

Let $C$ be a homotopical category.

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

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

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

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

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

• A left deformation retract for $F$ is a left deformation retract $C_Q$ of $C$ such that $F$ becomes a homotopical functor when restricted to $C_Q$.

Right deformations are defined analogously.

Generalization

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

• A deformation retract for a two-variable adjunction $(\otimes , hom_l, hom_r) : C \times D \to E$ consists of left deformation retracts $C_Q$, $D_Q$ for $C$ and $D$, respectively, and a right deformation retract $E_Q$ of $E$, such that

  • $\otimes$ is homotopical on $C_Q \times D_Q$;

  • $hom_l$ is homotopical on $C_Q^{op} \times E_Q$;

  • $hom_r$ is homotopical on $D_Q^{op} \times E_Q$.

References

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)