nLab categorical algebra




Broadly speaking, categorical algebra is algebra seen from and generalized via the point of view of category theory. Thus it studies those aspects of categorical and category-like constructions which are in the spirit of pure algebra.

First and foremost this includes the study of monoidal category theory, and the corresponding internal notions of monoid objects, module objects, etc.

More generally, it is about the study of

  1. algebras over\, algebraic theories,

  2. algebras over\, monads,

  3. algebras over\, operads.

An account of the basics may be found at geometry of physics – categories and toposes in the section Basic notions of categorical algebra.


Some references use “categorical algebra” much as a synonym for category theory as such:

Discussion more focused on actual universal algebra:

Exposition of basics of monoidal categories and categorical algebra:

See also

Last revised on August 12, 2023 at 11:32:48. See the history of this page for a list of all contributions to it.