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


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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


In topological spaces


For f,g:XYf,g\colon X \longrightarrow Y two continuous functions between topological spaces X,YX,Y, then a left homotopy

η:f Lg \eta \colon f \,\Rightarrow_L\, g

is a continuous function

η:X×IY \eta \;\colon\; X \times I \longrightarrow Y

out of the standard cylinder object over XX: the product space of XX with the Euclidean closed interval I[0,1]I \coloneqq [0,1], such that this fits into a commuting diagram of the form

X (id,δ 0) f X×I η Y (id,δ 1) g X. \array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,.

(graphics grabbed from J. Tauber here)


Let XX be a topological space and let x,yXx,y \in X be two of its points, regarded as functions x,y:*Xx,y \colon \ast \longrightarrow X from the point to XX. Then a left homotopy, def. 1, between these two functions is a commuting diagram of the form

* δ 0 x I η Y δ 1 y *. \array{ \ast \\ {}^{\mathllap{\delta_0}}\downarrow & \searrow^{\mathrlap{x}} \\ I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{\delta_1}}\uparrow & \nearrow_{\mathrlap{y}} \\ \ast } \,.

This is simply a continuous path in XX whose endpoints are xx and yy.

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.

In model categories

If 𝒞\mathcal{C} 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 𝒞\mathcal{C} be a model category and X𝒞X \in \mathcal{C} an object.

  • A path object Path(X)Path(X) for XX is a factorization of the diagonal X:XX×X\nabla_X \colon X \to X \times X as
X:XWiPath(X)(p 0,p 1)X×X. \nabla_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X \,.

where XPath(X)X\to Path(X) is a weak equivalence. This is called a good path object if in addition Path(X)X×XPath(X) \to X \times X is a fibration.

  • A cylinder object Cyl(X)Cyl(X) for XX is a factorization of the codiagonal (or “fold map”) Δ XXXX\Delta_X X \sqcup X \to X as
Δ X:XX(i 0,i 1)Cyl(X)pWX. \Delta_X \;\colon\; X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X \,.

where Cyl(X)XCyl(X) \to X is a weak equivalence. This is called a good cylinder object if in addition XXCyl(X)X \sqcup X \to Cyl(X) 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. 2. 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 X×[0,1]X\times [0,1] is a cylinder object, but not a good cylinder unless XX itself is cofibrant (a cell complex in this case).

More generally, the path object Path(X)Path(X) of def. 2 is analogous to the powering (I,X)\pitchfork(I,X) with an interval object and the cyclinder object Cyl(X)Cyl(X) is analogous to the tensoring with a cylinder object IXI\odot X. In fact, if 𝒞\mathcal{C} is a VV-enriched model category and XX 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 f,g:XYf,g \colon X \longrightarrow Y be two parallel morphisms in a model category.

  • A left homotopy η:f Lg\eta \colon f \Rightarrow_L g is a morphism η:Cyl(X)Y\eta \colon Cyl(X) \longrightarrow Y from a cylinder object of XX, def. 2, such that it makes this diagram commute:
X Cyl(X) X f η g Y. \array{ X &\longrightarrow& Cyl(X) &\longleftarrow& X \\ & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}} \\ && Y } \,.
  • A right homotopy η:f Rg\eta \colon f \Rightarrow_R g is a morphism η:XPath(Y)\eta \colon X \to Path(Y) to some path object of XX, def. 2, such that this diagram commutes:
X f η g Y Path(Y) Y. \array{ && X \\ & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}} \\ Y &\longleftarrow& Path(Y) &\longrightarrow& Y } \,.

By remark 1 it follows that in a TopTop-enriched 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.

For more see at homotopy in a model category.

In (co-)fibration categories

Clearly the concepf of left homotopy in def. 2 only needs part of the model category axioms and thus makes sense more generally in suitable cofibration categories. Dually, the concepf of path ojects in def. 2 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.

[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 at model category.

Revised on July 18, 2017 00:42:14 by Urs Schreiber (