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 ...
, more 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 ...
, internal categories are a generalisation of the notion of
small category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
, and are defined with respect to a fixed
ambient category
Ambient or Ambiance or Ambience may refer to:
Music and sound
* Ambience (sound recording), also known as atmospheres or backgrounds
* Ambient music, a genre of music that puts an emphasis on tone and atmosphere
* ''Ambient'' (album), by Moby
* ...
. If the ambient category is taken to be 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 of m ...
then one recovers the theory of small categories. In general, internal categories consist of a pair of objects in the ambient category—thought of as the 'object of objects' and 'object of morphisms'—together with a collection of morphisms in the ambient category satisfying certain identities.
Group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
s, are common examples of internal categories.
There are notions of internal
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s and
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s that make the collection of internal categories in a fixed category into a
2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of catego ...
.
Definitions
Let
be a category with
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
s. An internal category in
consists of the following data: two
-objects
named "object of objects" and "object of morphisms" respectively and four
-arrows
subject to coherence conditions expressing the axioms of category theory. See
.
See also
*
Enriched category In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the ho ...
References
*
Category theory
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