# Lifting property

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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.

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## 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,

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.

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

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## 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 , where 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 , and that right-lifting is also a kind of negation. Hence the classes obtained from by taking orthogonals an odd number of times, such as etc., represent various kinds of negation of , so each consists of morphisms which are far from having property .

### 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.^{[1]}

- 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, .^{[2]}

- Let be the category of chain complexes over a commutative ring . Let be the class of maps of form

- and be
- Then are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.
^{[3]}

### Elementary examples in various categories

In **Set**,

- is the class of surjections,

- is the class of injections.

In the category of modules over a commutative ring ,

- is the class of surjections, resp. injections,

- A module is projective, resp. injective, iff is in , resp. is in .

In the category of groups,

- , resp. , is the class of injections, resp. surjections (where denotes the infinite cyclic group),

- A group is a free group iff is in

- A group is torsion-free iff is in

For a finite group ,

- iff the order of is prime to ,

- 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 , resp. denote the discrete, resp. antidiscrete space with two points 0 and 1. Let denote the Sierpinski space of two points where the point 0 is open and the point 1 is closed, and let etc. denote the obvious embeddings.

- a space satisfies the separation axiom T
_{0}iff is in

- a space satisfies the separation axiom T
_{1}iff is in

- 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 spaces 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

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Source: "Lifting property", Wikipedia, Wikimedia Foundation, (2023, January 25th), https://en.wikipedia.org/wiki/Lifting_property.

## Further Reading

###### Notes

**^**Hovey, Mark.*Model Categories*. Def. 2.4.3, Th.2.4.9**^**Hovey, Mark.*Model Categories*. Def. 3.2.1, Th.3.6.5**^**Hovey, Mark.*Model Categories*. Def. 2.3.3, Th.2.3.11

###### References

- Hovey, Mark (1999).
*Model Categories*.

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