A morphismi in a category has the left lifting propertywith respect to a morphism p, and p also has the right lifting property with respect to i, sometimes denoted or, iff the following implication holds for each morphism f and g in the category:
if the outer square of the following diagram commutes, then there exists h completing the diagram, i.e. for each and such that there exists such that and.
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 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 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 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 and composition of morphisms, and contains all isomorphisms of C. Meanwhile, is closed under retracts, pushouts, coproducts and transfinite composition of morphisms, 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, 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 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 etc., represent various kinds of negation of C, 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.
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,.
In the category Top 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 X satisfies the separation axiom T0 iff is in
a space X satisfies the separation axiom T1 iff is in
is the class of maps such that the topology on A is the pullback of topology on B, i.e. the topology on A is the topology with least number of open sets such that the map is continuous,
is the class of maps such that the preimage of a connected closed open subset of Y is a connected closed open subset of X, e.g. X is connected iff is in,
a space X is Hausdorff iff for any injective map, it holds where denotes the three-point space with two open points a and b, and a closed pointx,
a space X is perfectly normal iff where the open interval goes to x, and maps to the point, and maps to the point, and denotes the three-point space with two closed points and one open point x.
A spaceX 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,
