E-M-factorization-system-in-a-category
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
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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
![]() |
such that
![]() |
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
.
Generalizations of factorization theory to flows and flows with respect to a functor can be found in [a5] and [a11], respectively.
References
[a1] | J. Adamek, H. Herrlich, G.E. Strecker, "Abstract and concrete categories" , Wiley–Interscience (1990) |
[a2] | A.K. Bousfield, "Constructions of factorization systems in categories" J. Pure Appl. Algebra , 9 (1977) pp. 207–220 |
[a3] | C. Cassidy, M. Hébert, G.M. Kelly, "Reflective subcategories, localizations and factorization systems" J. Austral. Math. Soc. , 38 (1985) pp. 287–329 (Corrigenda: 41 (1986), 286) |
[a4] | P.J. Freyd, G.M. Kelly, "Categories of continuous functors I" J. Pure Appl. Algebra , 2 (1972) pp. 169–191 |
[a5] | H. Herrlich, W. Meyer, "Factorization of flows and completeness of categories" Quaest. Math. , 17 : 1 (1994) pp. 1–11 |
[a6] | H. Herrlich, G. Salicrup, R. Vazquez, "Dispersed factorization structures" Canad. J. Math. , 31 (1979) pp. 1059–1071 |
[a7] | H. Herrlich, G.E. Strecker, "Semi-universal maps and universal initial completions" Pacific J. Math. , 82 (1979) pp. 407–428 |
[a8] | R.-E. Hoffmann, "Factorization of cones" Math. Nachr. , 87 (1979) pp. 221–238 |
[a9] | R. Nakagawa, "A note on ![]() |
[a10] | G.E. Strecker, "Perfect sources" A. Dold (ed.) B. Eckmann (ed.) , Categorical Topol. Proc. Conf. Mannheim 1975 , Lecture Notes Math. , 540 , Springer (1976) pp. 605–624 |
[a11] | G.E. Strecker, "Flows with respect to a functor" Appl. Categorical Struct. (to appear) |
[a12] | W. Tholen, "Factorizations of cones along a functor" Quaest. Math. , 2 (1977) pp. 335–353 |
[a13] | W. Tholen, "Factorizations, localizations, and the orthogonal subcategory problem" Math. Nachr. , 114 (1983) pp. 63–85 |
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