This page is about homotopy as a transformation. For homotopy sets in homotopy categories, see homotopy (as an operation).

Contents

Idea

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 $(\mathrm{\infty },1)$-category, then homotopies $f\sim g$ present the 2-cells $f\Rightarrow g$ in the resulting $(\mathrm{\infty },1)$-category.

Definition in enriched categories

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 $\mathrm{Top}$ such that $H(0)=f$ and $H(1)=g$. In $\mathrm{Top}$ 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 $X\to ⋔([0,1],Y)$.

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.

Definition in model categories

If $C$ is instead a model category, it has an intrinsic notion of homotopy determined by its factorizations.

• A path object $\mathrm{Path}(X)$ for an object $X$ is a factorization of the diagonal $X\to X\times X$ as

  $$X\to \mathrm{Path}(X)\to X\times X.$$X \to Path(X) \to X \times X \,.

  where $X\to \mathrm{Path}(X)$ is a weak equivalence.

• A cylinder object $\mathrm{Cyl}(X)$ is a factorization of the codiagonal (or “fold”) $X\bigsqcup X\to X$ as

  $$X\bigsqcup X\to \mathrm{Cyl}(X)\to X.$$X \sqcup X \to Cyl(X) \to X \,.

  where $\mathrm{Cyl}(X)\to X$ is a weak equivalence.

Frequently one asks as well that $\mathrm{Path}(X)\to X\times X$ be a fibration and $X\bigsqcup X\to \mathrm{Cyl}(X)$ 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 $\mathrm{Path}(X)$ as an analogue of $⋔(I,X)$ and $\mathrm{Cyl}(X)$ as an analogue of $I\odot X$. In fact, if $C$ is a $\mathrm{Top}$-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 $e\bigsqcup e\to I\to e$ 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 $\eta :\mathrm{Cyl}(X)\to Y$ such that

  $$\begin{array}{ccccc}X& \to & \mathrm{Cyl}(X)& \leftarrow & X\\ & {}_f\searrow & {\downarrow }^{\eta }& {\swarrow }_g\\ & & Y\end{array}.$$\array{ X &\rightarrow& Cyl(X) &\leftarrow& X \\ & {}_f\searrow &\downarrow^\eta& \swarrow_g \\ && Y } \,.

• A right homotopy between two morphisms $f,g:X\to Y$ in $C$ is a morphism $\eta :X\to \mathrm{Path}(Y)$ such that

  $$\begin{array}{ccc}& & X\\ & {}^f\swarrow & {\downarrow }^{\eta }& {\searrow }^g\\ Y& \leftarrow & \mathrm{Path}(Y)& \to & Y\end{array}.$$\array{ && X \\ & {}^f\swarrow & \downarrow^\eta & \searrow^{g} \\ Y &\leftarrow& Path(Y) &\rightarrow& Y } \,.

By the above remarks about powers and copowers, it follows that in a $\mathrm{Top}$-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.

Remarks

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.

Related concepts

• homotopy class

• higher homotopy

• transfor

  • natural transformation

  • pseudonatural transformation

  • lax natural transformation

• function extensionality

References

See the references at homotopy theory and model category.