This page is about homotopy as a transformation. For homotopy sets in homotopy categories, see homotopy (as an operation).
Paths and cylinders
In many categories in which one does homotopy theory, there is a notion of homotopy between morphisms, which is closely related to the higher morphisms in higher category theory. If we regard such a category as a presentation of an -category, then homotopies present the 2-cells in the resulting -category.
In enriched categories
If is enriched over Top, then a homotopy in between maps is a map in such that and . In itself this is the classical notion.
If has copowers, then an equivalent definition is a map , while if it has powers, an equivalent definition is a map .
There is a similar definition in a simplicially enriched category, replacing with the 1-simplex , with the caveat that in this case not all simplicial homotopies need be composable even if they match correctly. (This depends on whether or not all (2,1)-horns in the simplicial set, , have fillers.) Likewise in a dg-category we can use the “chain complex interval” to get a notion of chain homotopy.
In model categories
If is a model category, it has an intrinsic notion of homotopy determined by its factorizations. For more on the following see at homotopy in a model category.
Let be a model category and an object.
- A path object for is a factorization of the diagonal as
where is a weak equivalence. This is called a good path object if in addition is a fibration.
- A cylinder object for is a factorization of the codiagonal (or “fold map”) as
where is a weak equivalence. This is called a good cylinder object if in addition is a cofibration.
By the factorization axioms every object in a model category has both a good path object and as well as a good cylinder object according to def. 1. But in some situations one is genuinely interested in using non-good such objects.
For instance in the classical model structure on topological spaces, the obvious object is a cylinder object, but not a good cylinder unless itself is cofibrant (a cell complex in this case).
More generally, the path object of def. 1 is analogous to the powering with an interval object and the cyclinder object is analogous to the tensoring with a cylinder object . In fact, if is a -enriched model category and is fibrant/cofibrant, then these powers and copowers are in fact examples of (good) path and cylinder objects if the interval object is sufficiently good.
Let be two parallel morphisms in a model category.
- A left homotopy is a morphism from a cylinder object of , def. 1, such that it makes this diagram commute:
- A right homotopy is a morphism to some path object of , def. 1, such that this diagram commutes:
For more see at homotopy in a model category.
In (co-)fibration categories
Clearly the concepf of left homotopy in def. 1 only needs part of the model category axioms and thus makes sense more generally in suitable cofibration categories. Dually, the concepf of path ojects in def. 1 makes sense more generally in suitable fibration categories such as categories of fibrant objects in the sense of Brown.
Likewise if there is a cylinder functor, one gets functorially defined cylinder objects, etc.
See the references at homotopy theory and at model category.