This page is about homotopy as a transformation. For homotopy sets in homotopy categories, see homotopy (as an operation).
In many categories $C$ 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 $(\infty,1)$-category, then homotopies $f\sim g$ present the 2-cells $f\Rightarrow g$ in the resulting $(\infty,1)$-category.
If $C$ is enriched over Top, then a homotopy in $C$ between maps $f,g:X\,\rightrightarrows \,Y$ is a map $H:[0,1] \to C(X,Y)$ in $Top$ such that $H(0)=f$ and $H(1)=g$. In $Top$ itself this is the classical notion.
If $C$ has copowers, then an equivalent definition is a map $[0,1]\odot X\to Y$, while if it has powers, an equivalent definition is a map $X\to \pitchfork([0,1],Y)$.
There is a similar definition in a simplicially enriched category, replacing $[0,1]$ with the 1-simplex $\Delta^1$, 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, $C(X,Y)$, have fillers.) Likewise in a dg-category we can use the “chain complex interval” to get a notion of chain homotopy.
If $C$ is instead a model category, it has an intrinsic notion of homotopy determined by its factorizations.
A path object $Path(X)$ for an object $X$ is a factorization of the diagonal $X \to X \times X$ as
where $X\to Path(X)$ is a weak equivalence.
A cylinder object $Cyl(X)$ is a factorization of the codiagonal (or “fold”) $X \sqcup X \to X$ as
where $Cyl(X) \to X$ is a weak equivalence.
Frequently one asks as well that $Path(X)\to X\times X$ be a fibration and $X\sqcup X\to Cyl(X)$ be a cofibration; we call such paths and cylinders good. Clearly any object has a good path object and a good cylinder object. However, in the usual model structure on topological spaces, the obvious object $X\times I$ is a cylinder, but not a good cylinder unless $X$ itself is cofibrant.
We think of $Path(X)$ as an analogue of $\pitchfork(I,X)$ and $Cyl(X)$ as an analogue of $I\odot X$. In fact, if $C$ is a $Top$-enriched model category and $X$ is cofibrant, then these powers and copowers are in fact examples of path and cylinder objects. (This works more generally if $C$ is a $V$-model category and $e\sqcup e \to I \to e$ is a good cylinder object for the cofibrant unit object $e$ of $V$.)
Then:
A left homotopy between two morphisms $f,g : X \to Y$ in $C$ is a morphism $\eta : Cyl(X) \to Y$ such that
A right homotopy between two morphisms $f,g : X \to Y$ in $C$ is a morphism $\eta : X \to Path(Y)$ such that
By the above remarks about powers and copowers, it follows that in a $Top$-model category, any enriched homotopy between maps $X\to Y$ is a left homotopy if $X$ is cofibrant and a right homotopy if $Y$ is fibrant. Similar remarks hold for other enrichments.
Path objects and right homotopies also exist in various other situations, for instance, if there is not the full structure of a model category but just of a category of fibrant objects is given. Likewise for cylinder objects and left homotopies in a category of cofibrant objects.
Likewise if there is a cylinder functor, one gets functorially defined cylinder objects.
homotopy | cohomology | homology | |
---|---|---|---|
$[S^n,-]$ | $[-,A]$ | $(-) \otimes A$ | |
category theory | covariant hom | contravariant hom | tensor product |
homological algebra | Ext | Ext | Tor |
enriched category theory | end | end | coend |
homotopy theory | derived hom space $\mathbb{R}Hom(S^n,-)$ | cocycles $\mathbb{R}Hom(-,A)$ | derived tensor product $(-) \otimes^{\mathbb{L}} A$ |
See the references at homotopy theory and model category.