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 given class of morphisms.

It appears in a prominent way in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen.

It is also used in the definition of a factorization system, and of a weak factorization system, notions related to but less restrictive than the notion of a model category.

Several elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.

in a category has the left lifting property with respect to a morphism

also has the right lifting property with respect to

, iff the following implication holds for each morphism

; however, this can also refer to the stronger property that whenever

of morphisms in a category, its left orthogonal

, is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class

In notation, Taking the orthogonal of a class

is a simple way to define a class of morphisms excluding non-isomorphisms from

, in a way which is useful in a diagram chasing computation.

the simplest non-injection, are both precisely the class of injections, It is clear that

is always closed under retracts, pullbacks, (small) products (whenever they exist in the category) & composition of morphisms, and contains all isomorphisms (that is, invertible morphisms) of the underlying category.

is closed under retracts, pushouts, (small) coproducts & transfinite composition (filtered colimits) of morphisms (whenever they exist in the category), and also contains all isomorphisms.

A number of notions can be defined by passing to the left or right orthogonal several times starting from a list of explicit examples, i.e. as

is a class consisting of several explicitly given morphisms.

A useful intuition is to think that the property of left-lifting against a class

is a kind of negation of the property of being in

by taking orthogonals an odd number of times, such as

each consists of morphisms which are far from having property

has the path lifting property iff

has the homotopy lifting property iff

of modules over a commutative ring

denote the discrete, resp.

antidiscrete space with two points 0 and 1.

denote the obvious embeddings.

In the category of metric spaces with uniformly continuous maps.