homotopy lifting property


Homotopy theory

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

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see also algebraic topology



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Basic facts




In classical homotopy theory, a homotopy lifting property is a condition satisfied by continuous maps between topological spaces. More generally, such a condition may appear in more general context of a category with products and with an interval object II:

The Eckmann–Hilton dual of the homotopy lifting property is the homotopy extension property.


Let CC be a category with products and with interval object II.

A morphism EBE \to B has the homotopy lifting property if it has the right lifting property with respect to all morphisms of the form (Id,0):YY×I(Id,0) : Y \to Y \times I.

This means that for all commuting squares

Y f E p Y×I F B \array{ Y &\stackrel{f}\to& E \\ \downarrow && \downarrow^p \\ Y\times I &\stackrel{F}{\to}& B }

there exists a morphism σ:Y×IE\sigma : Y \times I \to E such that both triangles in

Y f E σ p Y×I F B \array{ Y &\stackrel{f}\to& E \\ \downarrow &{}^\sigma\nearrow& \downarrow^p \\ Y\times I &\stackrel{F}{\to}& B }


For Y=*Y = * a generator this can be rephrased as saying that the universal morphism E IB I× BEE^I \to B^I \times_B E induced by the commuting square

E I E B I B \array{ E^I &\to& E \\ \downarrow && \downarrow \\ B^I &\to& B }

is an epimorphism. If it is even an isomorphism then the lift σ\sigma exists uniquely .

The homotopy lifting property is an instance of a right lifting property.


For topological spaces

Here the ambient category is C=C = Top and the interval object is the topological interval I=[0,1]I = [0,1].

A continuous map p:EBp:E\to B of topological spaces satisfies the homotopy lifting property (or covering homotopy property) with respect to a space YY if for every commuting square in TopTop

Y f E σ 0 F˜ p Y×I F B. \array{ Y &\stackrel{f}\to& E \\ \downarrow^{\sigma_0} &{}^{\tilde{F}}\nearrow& \downarrow^p \\ Y\times I &\stackrel{F}{\to}& B } \,.

there is a diagonal such that the entire diagram commutes. The map σ 0:YY×I\sigma_0:Y\to Y\times I is given by y(y,0)y\mapsto (y,0) for yYy\in Y.

A map pp is a Hurewicz fibration if it satisfies the homotopy lifting property with respect to all spaces XX. A map pp is a Serre fibration if it satisfies the homotopy lifting property with respect to all disks (equivalently, all topological cubes).

There are weaker notions than the usual homotopy lifting property. For example, in the notion of Dold fibration one requires in the above diagram that the lower triangle is commutative while the upper one is commutative only up to a homotopy. Alternatively, one can characterize the Dold fibrations by the delayed homotopy lifting property, where instead the notion of delayed homotopy is used, but the lift makes the division of the square strictly commutative.

For metric spaces

For metric spaces, there is also a weaker notion, the approximate homotopy lifting property.

For quasi-categories

Morphism between quasi-categories that are left fibrations of quasi-categories satisfy the homotopy lifting property with respect to Δ[0]Δ[1]\Delta[0] \hookrightarrow \Delta[1]

For smooth bundles

Revised on July 12, 2017 16:43:09 by Urs Schreiber (