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.
Let $C$ be a homotopical category.
A left deformation of $C$ is a functor $Q:C\to C$ equipped with a natural weak equivalence
(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.
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 $(\otimes ,{\mathrm{hom}}_{l},{\mathrm{hom}}_{r}):C\times D\to E$ 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 ${C}_{Q}\times {D}_{Q}$;
${\mathrm{hom}}_{l}$ is homotopical on ${C}_{Q}^{\mathrm{op}}\times {E}_{Q}$;
${\mathrm{hom}}_{r}$ is homotopical on ${D}_{Q}^{\mathrm{op}}\times {E}_{Q}$.
The definition of deformation and deformation retract is in paragraph 40 of
The notion of deformation retract of a two-variable adjunction is definition 15.1, p. 43 in