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Internal category

Generalization of a small category

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated April 6, 2026

In mathematics, more specifically in category theory, internal categories are a generalization of the notion of a 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.

01Definitions

Let C be a category with pullbacks. An internal category in C consists of the following data: two C-objects C_{0},C_{1} named "object of objects" and "object of morphisms" respectively and four C-arrows d_{0},d_{1}:C_{1}\rightarrow C_{0},e:C_{0}\rightarrow C_{1},m:C_{1}\times _{C_{0}}C_{1}\rightarrow C_{1} subject to coherence conditions expressing the axioms of category theory. See .

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This article is adapted from the Wikipedia article “Internal category”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Enriched category

In category theory, a branch of mathematics, an enriched category generalizes the idea of a locally small category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms.

Double category

In mathematics, especially category theory, a double category is a generalization of a category where instead of morphisms, we have vertical morphisms, horizontal morphisms and 2-morphisms. Introduced by Ehresmann in 1960s, the notion may be compared with that of a bicategory; namely, the notion of a bicategory is obtained by enrichment, while the notion of a double category is obtained by internalization.