In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, an abstract mathematical discipline, a nodal decomposition of a morphism
is a representation of
as a product
, where
is a
strong epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
,
a
bimorphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
, and
a
strong monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphi ...
.
[A ]monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphism ...
is said to be strong, if for any epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f \ ...
and for any morphisms and such that there exists a morphism , such that and
Uniqueness and notations
If it exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions
and
there exist
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
s
and
such that
:
:
:
This property justifies some special notations for the elements of the nodal decomposition:
:
– here
and
are called the ''nodal coimage of
'',
and
the ''nodal image of
'', and
the ''nodal reduced part of
''.
In these notations the nodal decomposition takes the form
:
Connection with the basic decomposition in pre-abelian categories
In a
pre-abelian category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.
Spelled out in more detail, this means that a category C is pre-abelian if:
# C is preadditive, that is enric ...
each morphism
has a standard decomposition
:
,
called the ''basic decomposition'' (here
,
, and
are respectively the image, the coimage and the reduced part of the morphism
).
If a morphism
in a
pre-abelian category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.
Spelled out in more detail, this means that a category C is pre-abelian if:
# C is preadditive, that is enric ...
has a nodal decomposition, then there exist morphisms
and
which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities:
:
:
:
Categories with nodal decomposition
A category
is called a ''category with nodal decomposition'' if each morphism
has a nodal decomposition in
. This property plays an important role in constructing
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a shor ...
s and
refinement
Refinement may refer to: Mathematics
* Equilibrium refinement, the identification of actualized equilibria in game theory
* Refinement of an equivalence relation, in mathematics
** Refinement (topology), the refinement of an open cover in mathem ...
s in
.
In an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
the basic decomposition
:
is always nodal. As a corollary, ''all abelian categories have nodal decomposition''.
''If a
pre-abelian category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.
Spelled out in more detail, this means that a category C is pre-abelian if:
# C is preadditive, that is enric ...
is linearly complete,
[ A category is said to be ''linearly complete'', if any functor from a linearly ordered set into has ]direct
Direct may refer to:
Mathematics
* Directed set, in order theory
* Direct limit of (pre), sheaves
* Direct sum of modules, a construction in abstract algebra which combines several vector spaces
Computing
* Direct access (disambiguation), a ...
and inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
s. well-powered in strong monomorphisms
[A category is said to be ''well-powered in strong monomorphisms'', if for each object the category of all ]strong monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphi ...
s into is skeletally small (i.e. has a skeleton which is a set). and co-well-powered in strong epimorphisms,
[A category is said to be ''co-well-powered in strong epimorphisms'', if for each object the category of all ]strong epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
s from is skeletally small (i.e. has a skeleton which is a set). then
has nodal decomposition.''
More generally, ''suppose a category
is linearly complete,
well-powered in strong monomorphisms,
co-well-powered in strong epimorphisms,
and in addition strong epimorphisms discern monomorphisms
[It is said that ''strong epimorphisms discern monomorphisms'' in a category , if each morphism , which is not a monomorphism, can be represented as a composition , where is a ]strong epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
which is not an isomorphism. in
, and, dually, strong monomorphisms discern epimorphisms
[It is said that ''strong monomorphisms discern epimorphisms'' in a category , if each morphism , which is not an epimorphism, can be represented as a composition , where is a ]strong monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphi ...
which is not an isomorphism. in
, then
has nodal decomposition.''
The category Ste of
stereotype space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an iso ...
s (being non-abelian) has nodal decomposition, as well as the (non-
additive
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with f ...
) category SteAlg of
stereotype algebra In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra A over a topological field K is a ...
s .
Notes
References
*
*
*{{cite journal, last=Akbarov, first=S.S., title=Envelopes and refinements in categories, with applications to functional analysis, url=https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513, journal=Dissertationes Mathematicae, year=2016, volume=513, pages=1–188, arxiv=1110.2013, doi=10.4064/dm702-12-2015, s2cid=118895911
Category theory