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



In many categories CC 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 (,1)(\infty,1)-category, then homotopies fgf\sim g present the 2-cells fgf\Rightarrow g in the resulting (,1)(\infty,1)-category.

Definition in enriched categories

If CC is enriched over Top, then a homotopy in CC between maps f,g:XYf,g:X\,\rightrightarrows \,Y is a map H:[0,1]C(X,Y)H:[0,1] \to C(X,Y) in TopTop such that H(0)=fH(0)=f and H(1)=gH(1)=g. In TopTop itself this is the classical notion.

If CC has copowers, then an equivalent definition is a map [0,1]XY[0,1]\odot X\to Y, while if it has powers, an equivalent definition is a map X([0,1],Y)X\to \pitchfork([0,1],Y).

There is a similar definition in a simplicially enriched category, replacing [0,1][0,1] with the 1-simplex Δ 1\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)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 CC is instead a model category, it has an intrinsic notion of homotopy determined by its factorizations.

  • A path object Path(X)Path(X) for an object XX is a factorization of the diagonal XX×XX \to X \times X as

    XPath(X)X×X. X \to Path(X) \to X \times X \,.

    where XPath(X)X\to Path(X) is a weak equivalence.

  • A cylinder object Cyl(X)Cyl(X) is a factorization of the codiagonal (or “fold”) XXXX \sqcup X \to X as

    XXCyl(X)X. X \sqcup X \to Cyl(X) \to X \,.

    where Cyl(X)XCyl(X) \to X is a weak equivalence.

Frequently one asks as well that Path(X)X×XPath(X)\to X\times X be a fibration and XXCyl(X)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×IX\times I is a cylinder, but not a good cylinder unless XX itself is cofibrant.

We think of Path(X)Path(X) as an analogue of (I,X)\pitchfork(I,X) and Cyl(X)Cyl(X) as an analogue of IXI\odot X. In fact, if CC is a TopTop-enriched model category and XX is cofibrant, then these powers and copowers are in fact examples of path and cylinder objects. (This works more generally if CC is a VV-model category and eeIee\sqcup e \to I \to e is a good cylinder object for the cofibrant unit object ee of VV.)


  • A left homotopy between two morphisms f,g:XYf,g : X \to Y in CC is a morphism η:Cyl(X)Y\eta : Cyl(X) \to Y such that

    X Cyl(X) X f η g Y. \array{ X &\rightarrow& Cyl(X) &\leftarrow& X \\ & {}_f\searrow &\downarrow^\eta& \swarrow_g \\ && Y } \,.
  • A right homotopy between two morphisms f,g:XYf,g : X \to Y in CC is a morphism η:XPath(Y)\eta : X \to Path(Y) such that

    X f η g Y Path(Y) Y. \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 TopTop-model category, any enriched homotopy between maps XYX\to Y is a left homotopy if XX is cofibrant and a right homotopy if YY 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.

[S n,][S^n,-][,A][-,A]()A(-) \otimes A
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space Hom(S n,)\mathbb{R}Hom(S^n,-)cocycles Hom(,A)\mathbb{R}Hom(-,A)derived tensor product () 𝕃A(-) \otimes^{\mathbb{L}} A


See the references at homotopy theory and model category.

Revised on January 6, 2013 05:49:00 by Urs Schreiber (