In
mathematics, in particular in
category theory, the lifting property is a property of a pair of
morphisms in a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
. It is used in
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...
within
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
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
In mathematics, particularly in homotopy theory, a model category is a category theory, category with distinguished classes of morphisms ('arrows') called 'weak equivalence (homotopy theory), weak equivalences', 'fibrations' and 'cofibrations' sati ...
, an axiomatic framework for
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...
introduced by
Daniel Quillen. It is also used in the definition of a
factorization system In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory.
Definition
A factoriza ...
, 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
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, (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
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