In mathematics, more specifically in category theory, internal categories are a generalisation of the notion of small category, and are defined with respect to a fixed ambient category.
If the ambient category is taken to be the category of sets 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 objects, are common examples of internal categories.
There are notions of internal functors and natural transformations that make the collection of internal categories in a fixed category into a 2-category.
be a category with pullbacks.
consists of the following data: two
subject to coherence conditions expressing the axioms of category theory.
This category theory-related article is a stub.