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.
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:
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.
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 (that is, if and are objects, , and is a retract of , then ), 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.
Let , , and be morphisms such that exists. Then:
These two properties are useful when the category is equipped with a weak factorisation system consisting of epimorphisms and monomorphisms.
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.
Let denote any fixed singleton set, such as , and let denote any fixed set with two elements, such as .
Let denote the zero module and for each -module , let and denote the two unique morphisms between and .
Let denote the infinite cyclic group of integers under addition.
For a finite group ,
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 not closed) and the set is closed (and not open), and let , etc. denote the obvious embeddings.
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 .
Fibrations and cofibrations.