Glossary of category theory

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

The notations and the conventions used throughout the article are: The theory of categories originated ... with the need to guide complicated calculations involving passage to the limit in the study of the qualitative leap from spaces to homotopical/homological objects.

But category theory does not rest content with mere classification in the spirit of Wolffian metaphysics (although a few of its practitioners may do so); rather it is the mutability of mathematically precise structures (by morphisms) which is the essential content of category theory.

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.