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. A number of elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.
Often it is useful to think of lifting properties as a expressing a kind of qualitative negation (“Quillen negation”): The morphisms with the left/right lifting property against those in a class tend to be characterized by properties opposite of those in . For example, a morphism in Sets is surjective iff it has the right lifting property against the archetypical non-surjective map , and injective iff it has either left or right lifting property against the archetypical non-injective map . (For more such examples see at separation axioms in terms of lifting properties.)
Definition
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 ‘’weakly orthogonal to’‘ the morphism ; however, ‘’orthogonal to’‘ will 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 weak orthogonal’‘ or its ‘’left Quillen negation’‘ with respect to the lifting property, respectively its ‘’right weak orthogonal’‘ and its ‘’right Quillen negation’‘ 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 Quillen negation of a class is a simple way to define a class of morphisms excluding isomorphisms from , in a way which is useful in a diagram chasing computation.
Thus, in the category Set of sets, the right Quillen negation of the simplest non-surjection is the class of surjections. The left and right Quillen negation 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) and composition of morphisms, and contains all isomorphisms of C. Meanwhile, 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 of lifting properties
Decyphering notation in most of the examples below leads to standard definitions or reformulations. The intution behind most examples below is that the class of morphisms consists of simple or archetypal examples related to the property defined.
consider 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 Serre fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations. Hovey, Model Categories, Def. 2.4.3, Th.2.4.9
Hurewicz fibrations of topological spaces
A map has the ‘’path lifting property’‘ iff where is the inclusion of one end point of the closed interval into the interval .
Let be the class of boundary inclusions , and let be the class of horn inclusions . Then the classes of Kan fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, . (Model Categories, Def. 3.2.1, Th.3.6.5)
Many elementary properties in general topology, such as compactness, being dense or open, can be expressed as iterated Quillen negation of morphisms of finite topological spaces in the category Top of topological spaces. This leads to a concise, if useless, notation for a number of properties. Items below use notation for morphisms of finite topological spaces defined in the page on separation axioms in terms of lifting properties, and some examples are explained there in detail.
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
In topological spaces
The following lifting properties are calculated in the category of (all) topological spaces.
Iterated lifting properties
is the class of surjections
is the class of maps where or
is the class of subsets, i.e. injective maps where the topology on is induced from
is the class of maps , arbitrary
is the class of maps which admit a section
consists of maps such that either or
is the class of maps of form where is discrete
is the class of maps such that each connected subset of intersects the image of ; for “nice” spaces it means that the map is surjective, where “nice” means that connected componets are both open and closed.
is the class of maps of form where denotes the disjoint union of and .
is the class of closed inclusions where is closed
is the class of open inclusions where is open
is the class of closed maps where the topology on is pulled back from
is the class of open maps where the topology on is pulled back from
is the class of maps with dense image
is the class of closed subsets , a closed subset of
is the class of open subsets , a open subset of
is the class of subsets such that is the intersection of open subsets containing
where is the terminal object, and is a situs associated with a model and a formula, and and are objects of combinatorial nature, in the category of simplicial objects in the category of filters.