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 examples.

Formal definition

A morphism in a category has the left lifting property with respect to a morphism, and also has the right lifting property with respect to, sometimes denoted or, iff the following implication holds for each morphism and in the category:

if the outer square of the following diagram commutes, then there exists completing the diagram, i.e. for each and such that there exists such that and.

This is sometimes also known as the morphism being orthogonal to the morphism ; however, this can also refer to
the stronger property that whenever and are as above, the diagonal morphism exists and is also required to be unique.
For a class of morphisms in a category, its left orthogonal or with respect to the lifting property, respectively its right orthogonal or, is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class. In notation,

Properties

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.
In the category Set of sets, the right orthogonal of the simplest non-surjection is the class of surjections. The left and right orthogonals of the simplest non-injection, are both precisely the class of injections,
It is clear that and. The class is always closed under retracts, pullbacks, products & composition of morphisms, and contains all isomorphisms of the underlying category. Meanwhile, is closed under retracts, pushouts, coproducts & transfinite composition of morphisms, and also contains all isomorphisms.
Let,, and be morphisms such that exists. Then:

If and is an epimorphism, then.
If and is a monomorphism, then.

These two properties are useful when the category is equipped with a weak factorisation system consisting of epimorphisms and monomorphisms.

Examples

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, etc., where is a given class of morphisms. A useful intuition is to think that the left and right lifting properties against a class are a way of expressing a negation of some property of the morphisms in. In this vein, performing a "double negation" can be seen as a kind of "closure" or "completion" procedure.

Elementary examples in various categories

In '''Set'''

Let denote any fixed singleton set, such as, and let denote any fixed set with two elements, such as.

If denotes either of the two functions from to, then is the class of surjections.
If is the unique function from to, then is the class of injections.

In the category of modules">Module (mathematics)">modules over a commutative ring R

Let denote the zero module and for each -module, let and denote the two unique morphisms between and.

is the class of surjective module homomorphisms.
is the class of injective module homomorphisms.
A module is projective if and only if is in.
A module is injective if and only if is in.

In the category of groups">Group (mathematics)">groups

Let denote the infinite cyclic group of integers under addition.

is the class of surjective group homomorphisms.
is the class of injective group homomorphisms.
A group is a free group if and only if is in.
A group is torsion-free if and only if is in.
A subgroup of a group is pure if and only if is in.

For a finite group,

iff the order of is prime to iff.
iff is a -group.
is nilpotent iff the diagonal map is in where denotes the class of maps.
a finite group is soluble iff is in

In the category of topological spaces

Let and denote a two-element set with the discrete topology and the indiscrete topology, respectively. Let denote the Sierpinski space of two points, in which the set is open and the set is closed, and let, etc. denote the obvious embeddings.

A space is a T₀ space if and only if is in.
A space is a T₁ space if and only if is in.
is the class of maps with dense image.
is the class of maps such that the topology on is the pullback of topology on, i.e. the topology on is the topology with least number of open sets such that the map is continuous,
is the class of surjective maps,
is the class of maps of form where is discrete,
is the class of maps such that each connected component of intersects,
is the class of injective maps,
is the class of maps such that the preimage of a connected closed open subset of is a connected closed open subset of, e.g. is connected iff is in,
for a connected space, each continuous function on is bounded iff where is the map from the disjoint union of open intervals into the real line
a space is Hausdorff iff for any injective map, it holds where denotes the three-point space with two open points and, and a closed point,
a space is perfectly normal iff where the open interval goes to, and maps to the point, and maps to the point, and denotes the three-point space with two closed points and one open point.

In the category of [metric space]s with [uniformly continuous] maps

A space is complete iff where is the obvious inclusion between the two subspaces of the real line with induced metric, and is the metric space consisting of a single point,
A subspace is closed iff

Examples of lifting properties in algebraic topology

A map has the path lifting property iff where is the inclusion of one end point of the closed interval into the interval.
A map has the homotopy lifting property iff where is the map.

Examples of lifting properties coming from model categories

Fibrations and cofibrations.

Let Top be the category of topological spaces, and let be the class of maps, embeddings of the boundary of a ball into the ball. Let be the class of maps embedding the upper semi-sphere into the disk. are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.
Let sSet be the category of simplicial sets. Let be the class of boundary inclusions, and let be the class of horn inclusions. Then the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively,.
Let be the category of chain complexes over a commutative ring. Let be the class of maps of form