This is a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.)
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:
[T]he issue of comparing definitions of weak n-category is a slippery one, as it is hard to say what it even means for two such definitions to be equivalent. [...] It is widely held that the structure formed by weak n-categories and the functors, transformations, ... between them should be a weak (n + 1)-category; and if this is the case then the question is whether your weak (n + 1)-category of weak n-categories is equivalent to mine—but whose definition of weak (n + 1)-category are we using here... ?
Tom Leinster, A survey of definitions of n-category
Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.
Barry Mazur, Thinking about Grothendieck
where Nat means the set of natural transformations. In particular, the functor