Glossary of category theory

This is a glossary of properties and concepts in category theory in mathematics.

Notes on foundations: In many expositions, the set-theoretic issues are ignored; this means, for instance, that one does not distinguish between small and large categories and that one can arbitrarily form a localization of a category. Like those expositions, this glossary also generally ignores the set-theoretic issues, except when they are relevant

Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also glossary of algebraic topology.
The notations and the conventions used throughout the article are:

=, which is viewed as a category
Cat, the category of categories, where the objects are categories and the morphisms functors.
Fct, the functor category: the category of functors from a category C to a category D.
Set, the category of sets.
sSet, the category of simplicial sets.
"weak" instead of "strict" is given the default status; e.g., "n-category" means "weak n-category", not the strict one, by default.
By an ∞-category, we mean a quasi-category, the most popular model, unless other models are being discussed.
The number zero 0 is a natural number.
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