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.
Formal definition.
A morphism "i" in a category has the "left lifting property" with respect to a morphism "p", and "p" also has the "right lifting property" with respect to "i", sometimes denoted formula_1 or formula_2, iff the following implication holds for each morphism "f" and "g" in the category:
This is sometimes also known as the morphism "i" being "orthogonal to" the morphism "p"; however, this can also refer to
the stronger property that whenever "f" and "g" are as above, the diagonal morphism "h" exists and is also required to be unique.
For a class "C" of morphisms in a category, its "left orthogonal" formula_9 or formula_10 with respect to the lifting property, respectively its "right orthogonal" formula_11 or formula_12, is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class "C". In notation,
Taking the orthogonal of a class "C" is a simple way to define a class of morphisms excluding non-isomorphisms from "C", in a way which is useful in a diagram chasing computation.
Thus, in the category Set of sets, the right orthogonal formula_14 of the simplest non-surjection formula_15 is the class of surjections. The left and right orthogonals of formula_16 the simplest non-injection, are both precisely the class of injections, 
It is clear that formula_18 and formula_19. The class formula_11 is always closed under retracts, pullbacks, (small) products (whenever they exist in the category) and composition of morphisms, and contains all isomorphisms of C. Meanwhile, formula_9 is closed under retracts, pushouts, (small) coproducts and transfinite composition (filtered colimits) of morphisms (whenever they exist in the category), and also contains all isomorphisms.
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 formula_22, where formula_23 is a class consisting of several explicitly given morphisms. A useful intuition is to think that the property of left-lifting against a class "C" is a kind of negation
of the property of being in "C", and that right-lifting is also a kind of negation. Hence the classes obtained from "C" by taking orthogonals an odd number of times, such as formula_24 etc., represent various kinds of negation of "C", so formula_24 each consists of morphisms which are far from having property formula_23.
Examples of lifting properties in algebraic topology.
A map formula_27 has the "path lifting property" iff formula_28 where formula_29 is the inclusion of one end point of the closed interval into the interval formula_30.
A map formula_27 has the homotopy lifting property iff formula_32 where formula_33 is the map formula_34.
Examples of lifting properties coming from model categories.
Fibrations and cofibrations.
Elementary examples in various categories.
In Set, 
In the category "R"-Mod of modules over a commutative ring "R",
In the category Grp of groups, 
For a finite group "G", 
In the category Top of topological spaces, let formula_75, resp. formula_76 denote the discrete, resp. antidiscrete space with two points 0 and 1. Let formula_77 denote the Sierpinski space of two points where the point 0 is open and the point 1 is closed, and let formula_78 etc. denote the obvious embeddings.
In the category of metric spaces with uniformly continuous maps.