In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically 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 ...
, a semi-abelian category is 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 ...
in which the induced
morphism
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 a ...
is 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 a ...
, i.e., 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 ...
and an
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 \ ...
, for every morphism
.
The history of the notion is intertwined with that of a
quasi-abelian category, as, for awhile, it was not known whether the two notions are distinct (see
quasi-abelian category#History).
Properties
The two properties used in the definition can be characterized by several equivalent conditions.
Every semi-abelian category has a
maximal exact structure.
If a semi-abelian category is not
quasi-abelian, then the class of all kernel-cokernel pairs does not form an
exact structure.
Examples
Every
quasiabelian category is semiabelian. In particular, every
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 ...
is semi-abelian. Non-quasiabelian examples are the following.
* The
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)
* ...
of (possibly non-
Hausdorff)
bornological spaces is semiabelian.
* Let
be the
quiver
A quiver is a container for holding arrows, bolts, ammo, projectiles, darts, or javelins. It can be carried on an archer's body, the bow, or the ground, depending on the type of shooting and the archer's personal preference. Quivers were trad ...
:and
be a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. The category of
finitely generated projective module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over the
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
is semiabelian.
Left and right semi-abelian categories
By dividing the two conditions on the induced map in the definition, one can define left semi-abelian categories by requiring that
is a monomorphism for each morphism
. Accordingly, right semi-abelian categories are pre-abelian categories such that
is an epimorphism for each morphism
.
If a category is left semi-abelian and
right quasi-abelian, then it is already quasi-abelian. The same holds, if the category is right semi-abelian and left quasi-abelian.
[Rump, 2001.]
Citations
References
*
José Bonet
José is a predominantly Spanish and Portuguese form of the given name Joseph. While spelled alike, this name is pronounced differently in each language: Spanish ; Portuguese (or ).
In French, the name ''José'', pronounced , is an old vernacul ...
, J.,
Susanne Dierolf, The pullback for bornological and ultrabornological spaces. Note Mat. 25(1), 63–67 (2005/2006).
* Yaroslav Kopylov and Sven-Ake Wegner, On the notion of a semi-abelian category in the sense of Palamodov, Appl. Categ. Structures 20 (5) (2012) 531–541.
* Wolfgang Rump, A counterexample to Raikov's conjecture, Bull. London Math. Soc. 40, 985–994 (2008).
* Wolfgang Rump, Almost abelian categories, Cahiers Topologie Géom. Différentielle Catég. 42(3), 163–225 (2001).
* Wolfgang Rump, Analysis of a problem of Raikov with applications to barreled and bornological spaces, J. Pure and Appl. Algebra 215, 44–52 (2011).
* Dennis Sieg and Sven-Ake Wegner, Maximal exact structures on additive categories, Math. Nachr. 284 (2011), 2093–2100.
Additive categories