In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in particular in
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
within
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the homotopy lifting property (also known as an instance of the
right lifting property
In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly give ...
or the covering homotopy axiom) is a technical condition on a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
from a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''E'' to another one, ''B''. It is designed to support the picture of ''E'' "above" ''B'' by allowing a
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
taking place in ''B'' to be moved "upstairs" to ''E''.
For example, a
covering map has a property of ''unique'' local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
s. The homotopy lifting property will hold in many situations, such as the projection in a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
,
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
or
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all ma ...
, where there need be no unique way of lifting.
Formal definition
Assume from now on all maps are continuous functions from one topological space to another. Given a map
, and a space
, one says that
has the homotopy lifting property,
[ page 7] or that
has the homotopy lifting property with respect to
, if:
*for any
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
, and
*for any map
lifting
(i.e., so that
),
there exists a homotopy
lifting
(i.e., so that
) which also satisfies
.
The following diagram depicts this situation:
The outer square (without the dotted arrow) commutes if and only if the hypotheses of the
lifting property
In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly give ...
are true. A lifting
corresponds to a dotted arrow making the diagram commute. This diagram is dual to that of the
homotopy extension property In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual ...
; this duality is loosely referred to as
Eckmann–Hilton duality
In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram for a particular concept and reversing the direction of all arrows, much as in cat ...
.
If the map
satisfies the homotopy lifting property with respect to ''all'' spaces ''X'', then
is called a
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all ma ...
, or one sometimes simply says that ''
has the homotopy lifting property''.
A weaker notion of fibration is
Serre fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all ma ...
, for which homotopy lifting is only required for all
CW complexes .
Generalization: homotopy lifting extension property
There is a common generalization of the homotopy lifting property and the
homotopy extension property In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual ...
. Given a pair of spaces
, for simplicity we denote