In
mathematics, specifically
category theory, a family of generators (or family of separators) of 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) ...
is a collection
of objects in
, such that for any two ''distinct''
morphisms
in
, that is with
, there is some
in
and some morphism
such that
If the collection consists of a single object
, we say it is a generator (or separator).
Generators are central to the definition of
Grothendieck categories.
The
dual concept is called a cogenerator or coseparator.
Examples
* In the category of
abelian groups
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
, the group of integers
is a generator: If ''f'' and ''g'' are different, then there is an element
, such that
. Hence the map
suffices.
* Similarly, the one-point
set is a generator for the
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
. In fact, any nonempty set is a generator.
* In the
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
, any set with at least two elements is a cogenerator.
* In the category of modules over a
ring ''R'', a generator in a finite direct sum with itself contains an isomorphic copy of ''R'' as a direct summand. Consequently, a generator module is faithful, i.e. has zero
annihilator.
References
* , p. 123, section V.7
External links
*
Category theory
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