homotopy extension property


Homotopy theory

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

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see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




A morphism is said to have the homotopy extension property if homotopies out of its domain extend to homotopies out of its codomain.

More in detail, given a category with a concept of path space object and hence of right homotopy, then a morphism i:ABi \colon A \longrightarrow B is said to have the right homotopy extension property with respect to an object XX if it has the left lifting property against one of the two path endpoint evaluation maps p 0:Path(X)Xp_0 \colon Path(X) \longrightarrow X, i.e. if in every commuting square as below, a diagonal lift exists:

A Path(X) i p 0 B X. \array{ A &\longrightarrow& Path(X) \\ {}^{\mathllap{i}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{p_0}} \\ B &\longrightarrow& X } \,.

Here the top morphism exhibits a right homotopy between two morphisms AXA\to X, and the diagonal filler is an extension of the top morphism along ii and exhibiting a compatible right homotopy between morphisms BXB\to X, whence the terminology.

The concept of homotopy extension property is the Eckmann-Hilton dual of that of (left) homotopy lifting property, where instead one considers the presence of a cylinder object, hence a notion of left homotopy, and lifting in diagrams of the form

A X ι 0 f Cyl(A) Y. \array{ A &\longrightarrow& X \\ {}^{\iota_0}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{f} \\ Cyl(A) &\longrightarrow& Y } \,.

In situations where both path space objects as well as cylinder objects exist and are compatible, so that the concepts of left and right homotopy coincide, one may equivalently rephrase right homotopy extension in terms of left homotopy (and left homotopy lifting in terms of right homotopy).

The resulting definition is necessarily less transparent (def. below), but it happens to be more commonly used in the literature. Specifically in the archetypical case that the ambient category is that of topological spaces and cylinders and path space objects are induced from the standard topological interval object, a Hurewicz cofibration (often just called cofibration ) is a continuous function that satisfies the left homotopy extension property with respect to all topological spaces.


In topological spaces


(homotopy extension property for left homotopies in topological spaces)
A continuous function i:AXi \colon A \xrightarrow{\;} X of topological spaces is said to satisfy the (left) homotopy extension property (HEP) with respect to a space YY if

  • for any map f:XYf \colon X \longrightarrow Y and any left homotopy η:A×[0,1]Y\eta \,\colon\, A \times [0,1] \longrightarrow Y such that η(,0)=fi\eta(-,0) = f \circ i, there exists a left homotopy η^:X×[0,1]Y\widehat{\eta} \,\colon\, X \times [0,1] \longrightarrow Y such that η^(i×id)=η\widehat{\eta} \circ (i \times id) = \eta;


  • given a commuting diagram in TopSp of solid arrows as shown below, there exists a dashed arrow η^\widehat \eta making all sub-diagrams commute:

The map ff is sometimes said to be the initial condition of a homotopy extension problem and η^\widehat{\eta} is called the extension of the homotopy η\eta subject to that initial condition.

The map ii is called a Hurewicz cofibration if it satisfies this homotopy extension property with respect to all spaces YY.


(re-formulation in terms of right homotopies)
In a convenient cartesian closed category TopSpTopSp of topological spaces (such as that of compactly generated topological spaces) the product \dashv internal hom-adjunction

TopSp() [0,1]()×[0,1]TopSp TopSp \underoverset {\underset{ (-)^{[0,1]} }{\longrightarrow}} {\overset{ (-) \times [0,1] }{\longleftarrow}} {\;\;\;\;\;\bot\;\;\;\;\;} TopSp

allows to pass to adjunct morphisms in Def.

TopSp(A×[0,1],Y) TopSp(A,Y [0,1]) η η \array{ TopSp \big( A \times [0,1] ,\, Y \big) &\simeq& TopSp \big( A ,\, Y^{[0,1]} \big) \\ \eta &\leftrightarrow& \eta' }

and equivalently express the above diagram of left homotopies as the following (somewhat more transparent) diagram of right homotopies, where ev 0:Y [0,1]Yev_0 \,\colon\, Y^{[0,1]}\to Y denotes evaluation at 0 γγ(0)\gamma \mapsto \gamma(0):

In this equivalent formulation, the homotopy extension property is simply the right lifting property against the evaluation map ev 0:Y [0,1]Yev_0 \;\colon\; Y^{[0,1]} \xrightarrow{\;} Y out of the path space of YY.

(e.g. May 1999, p. 43)




If a map i:AXi\colon A\to X has the homotopy extension property with respect to a space YY, then for any map g:AZg \colon A\to Z, the pushout g *(i)=i⨿ AZ:ZX⨿ AZg_*(i)=i\amalg_A Z :Z\to X\amalg_A Z has the homotopy extension property with respect to the space YY.

This is a general statement about classes of morphisms defined by a left lifting property, see at injective and projective morphisms – closure properties


We would like to find F˜\tilde{F} to complete the commutative diagram

A g Z f Y I i g *(i) F˜ ev 0 X i *(g) X⨿ AZ F Y. \array{ A &\stackrel{g}\to&Z&\stackrel{f}\to& Y^I \\ \downarrow^{i}&&\downarrow^{g_*(i)} &\nearrow {\tilde{F}}& \downarrow^{ev_0} \\ X&\stackrel{i_*(g)}\to&X\amalg_A Z&\stackrel{F}{\to}& Y } \,.

Consider the external square obtained by composing the horizontal arrows:

A fg Y I i G˜ ev 0 X Fi *(g) Y. \array{ A &\stackrel{f\circ g}\to& Y^I \\ \downarrow^{i} &\nearrow {\exists\tilde{G}}& \downarrow^{ev_0} \\ X&\stackrel{F\circ i_*(g)}{\to}& Y } \,.

By the assumption on ii, there is a G˜\tilde{G} as in the diagram, such that both triangles commute, i.e. ev 0G˜=Fi *(g)ev_0\circ\tilde{G}=F\circ i_*(g) and G˜i=F˜g\tilde{G}\circ i = \tilde{F}\circ g.

If i:AXi:A\to X is satisfying the HEP with respect to YY then there is a diagonal in that external square which is some map G˜:XY I\tilde{G}:X\to Y^I. This map together with f:ZY If:Z\to Y^I, by the universal property of pushout, determines a unique map F˜:X⨿ AZY I\tilde{F}:X\amalg_A Z\to Y^I such that F˜i *(g)=G˜\tilde{F}\circ i_*(g)=\tilde{G} and F˜g *(i)=f\tilde{F}\circ g_*(i)=f. We need to show only that ev 0F˜=Fev_0\circ\tilde{F}=F as F˜g *(i)=f\tilde{F}\circ g_*(i)=f holds by the construction of F˜\tilde{F} as stated.

By the definition of G˜\tilde{G} and the commutativity of the original double square diagram, ev 0F˜i *(g)=ev 0G˜=Fi *(g)ev_0\circ \tilde{F}\circ i_*(g)=ev_0\circ\tilde{G}=F\circ i_*(g) and ev 0F˜g *(i)=ev 0f=Fg *(i)ev_0\circ \tilde{F}\circ g_*(i)=ev_0\circ f=F\circ g_*(i). This is almost what we wanted except that we precompose the wanted identity with both maps into the pushout. Thus by the uniqueness part of the universal property of pushout it follows that ev 0F˜=Fev_0\circ\tilde{F}=F.


Textbook accounts:

See also:

The terminology “h-cofibration” is due to: