The simple fact that every function between sets can be factored through its image (i.e., written as a composite
where is the codomain-restriction of and is the inclusion) is abstracted in category theory to an axiomatic theory of factorization structures for morphisms of a category . Here, and are classes of -morphisms (the requirements and were originally included, but later dropped) such that each -morphism has an -factorization
Clearly, further assumptions on and are required to make the factorization theory useful. A careful analysis has revealed that the crucial requirement that causes -factorizations to have appropriate characteristics is the so-called "unique E,M-diagonalization" condition, described in 3) below. Such factorization structures for morphisms have turned out to be especially useful for "well-behaved" categories (e.g., those having products and satisfying suitable smallness conditions). Morphism factorizations have been transformed into powerful categorical tools by successive generalizations to
a) factorization structures for sources in a category;
b) factorization structures for structured sources with respect to a functor;
c) factorization structures for flows in a category; and
d) factorization structures for structured flows with respect to a functor. The simplest of these is described first and in most detail. A general reference for this area is [a1], Chaps. IV; V.
Let and be classes of morphisms in a category (cf. also Morphism). Then is called a factorization structure for morphisms in , and is called -structured, provided that
1) each of and is closed under composition with isomorphisms;
2) has -factorizations (of morphisms); i.e., each morphism in has a factorization , with and ; and
3) has the unique -diagonalization property; i.e., for each commutative square
with and , there exists a unique diagonal, i.e., a morphism such that and . For example, the category of sets and functions has exactly four different factorization structures for morphisms, the most frequently used of which is (surjections, injections) described above, whereas the category of topological spaces and continuous functions has more than a proper class of different factorization structures for morphisms (see [a6]), but is not one of them (since it does not satisfy the diagonalization condition).
Diagonalization is crucial in that it guarantees essential uniqueness of factorizations. Also, it can be shown that each of and determines the other via the diagonal property, that and are compositive, and that . Many other pleasant properties of and follow from the definition above. and are dual to each other, is well-behaved with respect to limits and is well-behaved with respect to co-limits. Also, there exist satisfactory external characterizations of classes in a category that guarantee the existence of a class such that will be a factorization system for morphisms (see, e.g., [a2]). Many familiar categories have particular morphism factorization structures. Every finitely-complete category that has intersections is -structured. Each category that has equalizers and intersections is -structured, and a category that has pullbacks and co-equalizers is -structured if and only if regular epimorphisms in it are compositive.
Factorization structures for sources (i.e., families of morphisms with a common domain) in a category are defined quite analogously to those for single morphisms. Here, one has a class of morphisms and a family of sources, each closed under composition with isomorphisms, such that each source in has a factorization with and , and each commuting square in , with sources as right side and bottom side, a member of as top side and a member of as bottom side, has a diagonalization. A category that has these properties is called an -category. Notice that now and are no longer dual. The dual theory is that of a factorization structure for sinks, i.e., an -category. Interestingly, in any -category, must be contained in the class of all epimorphisms. (As a consequence, uniqueness of the diagonal comes without hypothesizing it.) However, is contained in the family of all mono-sources if and only if has co-equalizers and contains all regular epimorphisms. There exist reasonable external characterizations of classes in a category that guarantee the existence of a family such that is an -category (see e.g., [a1], 15.14) and a reasonable theory exists for extending factorization structures for morphisms to those for sources (respectively, sinks).
Factorization structures with respect to functors provide yet a further generalization, as follows.
Let be a functor, let be a class of -structured arrows, and let be a conglomerate of -sources. is called a factorization structure for , and is called an -functor provided that:
A) and are closed under composition with isomorphisms;
B) has -factorizations, i.e., for each -structured source there exist
C) has the unique -diagonalization property, i.e., whenever and are -structured arrows with , and and are -sources with , such that for each , then there exists a unique diagonal, i.e., an -morphism with and .
Interestingly, this precisely captures the important categorical notion of adjointness: i.e., a functor is an adjoint functor if and only if it is an -functor for some and .
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E-M-factorization-system-in-a-category. G. Strecker (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=E-M-factorization-system-in-a-category&oldid=15835