nLab
homotopy (as a transformation)

• for disambiguation see homotopy

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 $(\mathrm{\infty }\mathrm{,1})$-category, then homotopies $f\sim g$ present the 2-cells $f\Rightarrow g$ in the resulting $(\mathrm{\infty }\mathrm{,1})$-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 $H:[\mathrm{0,1}]\to C(X,Y)$ 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 $[\mathrm{0,1}]\odot X\to Y$, while if it has powers, an equivalent definition is a map $X\to ⋔([\mathrm{0,1}],Y)$.

There is a similar definition in a simplicially enriched category, replacing $[\mathrm{0,1}]$ with the 1-simplex ${\Delta }^1$, with the caveat that in this case not all simplicial homotopies can be composed. 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 $V$.)

Are there any interesting consequences or conditions for the existence of an actual object $I$ that produces path objects and cylinder objects in that way?

One consequence of a well-behaved such object $I$ is the existence of model structures on categories of operads (Berger-Moerdijk 2003).

Urs: I need to look at Berger-Moerdijk. Just yesterday I wrote down a definition of “category with interval object” myself in an attempt to capture precisely this idea. I said:

Definition: a category $V$ with interval object is a closed monoidal homotopical category whose homotopical structure extends to a category of fibrant objects and eqipped with with $\sigma ,\tau :\mathrm{pt}\to I$ in $V$ an internal co-category such that for every object $B$ the object $[I,B]$ is a path object of $B$.

Examples are essentially all categories $V$ of higher fibrant structures, I think, Kan complexes, higher categories, etc. The interval object is always the obvious one in these cases. There is an obvious generalization to the non-fibrant case, too, I think.

And in a category with interval object one can do a bunch of things that one would want to do with higher structures – notably one can do nonabelian principal $\mathrm{\infty }$-bundles, I think, as I try to describe here.

Tim The question asked at the start of this query relates to a large part of the modelizer story which occupies quite a large part of Grothendieck’s Pursuit of Stacks. For a discussion and quite a detailed treatment of cylinder bjects and the way in which their properties influence what homotopical theorem hold in that context you might look at my book with Heiner Kamps.

Urs: may I ask that we move this discussion to interval object where it may have more room to develop in its own right, not just being an afterthought to this entry here. Tim: is maybe the answer to the question which I am asking at interval object in that book of yours which you mention? I will have a look.

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.

References

• W. G. Dwyer and J. Spalinski. “Homotopy Theories and Model Categories,” 1995.

• Clemens Berger and Ieke Moerdijk. “Axiomatic homotopy theory for operads,” 2003.

• K. H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory (GoogleBooks)