Paths and cylinders
A deformation retract is a retract which is also a section up to homotopy. Equivalently, it is a homotopy equivalence one of whose two homotopies is in fact an identity.
Let be a category equipped with a notion of homotopy between its morphisms. Then a deformation retraction of a morphism
(the deformation retract itself) is another morphism
In particular, if “homotopy” in means left homotopy with respect to an cylinder object
then a deformation retract of is a morphism such that and such that there exists a morphism fitting into a diagram
Hence a deformation retract is a (left) homotopy equivalence where one of the two homotopies occuring is in fact an identity.
If the cylinder object assignment here is functorial, we say that is a strong deformation retract if moreover
(hence if the homotopy restricted to the inclusion is “constant” as seen by the chosen cylinder object).
In parts of the literature, deformation retracts are required to be strong by default.
In topological spaces
In the category Top of topological spaces the standard cylinder object is given by cartesian product with the interval .
With respect to the corresponding notion of left homotopy, if is a topological space and a subspace, then is a strong deformation retract of if there exists a continuous map such that for all , , for all and for all .
Equivalently, there are continuous maps and such that and , where denotes homotopy with fixed . More generally, for any continuous map we say that it is deformation retractable if there is such that and .
A pair is an NDR-pair if there are two continuous maps, such that for all and all , for all , and for all such that . If is an NDR-pair, then the inclusion has a left homotopy inverse iff is also a retract of (in Top, in the standard categorical sense).
The pair is a DR-pair if it is a deformation retract and there is a function such that (i.e. it gives simultaneously a deformation retract and a NDR-pair). If is an NDR-pair then the inclusion is a homotopy equivalence iff is a deformation retract of . Any map is a homotopy equivalence iff is the deformation retract of the mapping cylinder of . If is an NDR-pair and is contractible, then the quotient map is a homotopy equivalence.