This page is about homotopy as a transformation. For homotopy sets in homotopy categories, see homotopy (as an operation).
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In many categories in which one does homotopy theory, there is a notion of homotopy between morphisms, which is closely related to the 2-morphisms in higher category theory: a homotopy between two morphisms is a way in which they are equivalent.
If we regard such a category as a presentation of an -category, then homotopies present the 2-cells in the resulting -category.
For two continuous functions between topological spaces , then a left homotopy
is a continuous function
out of the standard cylinder object over : the product space of with the Euclidean closed interval , such that this fits into a commuting diagram of the form
(graphics grabbed from J. Tauber here)
Let be a topological space and let be two of its points, regarded as functions from the point to . Then a left homotopy, def. , between these two functions is a commuting diagram of the form
This is simply a continuous path in whose endpoints are and .
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.
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.
where is a weak equivalence. This is called a good path object if in addition is a fibration.
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. . 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. is analogous to the powering with an interval object and the cylinder object is analogous to the tensoring with an interval 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.
By remark it follows that in a -enriched model category, any enriched homotopy between maps is a left homotopy if is cofibrant and a right homotopy if is fibrant. Similar remarks hold for other enrichments.
For more see at homotopy in a model category.
Clearly the concept of left homotopy in def. only needs part of the model category axioms and thus makes sense more generally in suitable cofibration categories. Dually, the concept of path objects in def. 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.
In dependent type theory, let be a type and let be a type family indexed by , and let be two elements of a dependent product type of a type family . The type of homotopies between and is the type
A homotopy between and is simply an element .
Note that a homotopy is not the same as an identification . However this can be made so if one assumes function extensionality.
See the references at homotopy theory and at model category.
Discussion in computational topology:
For homotopies in Martin-Löf dependent type theory:
Last revised on December 23, 2022 at 16:36:36. See the history of this page for a list of all contributions to it.