nLab categorical algebra

Idea

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.

Related concepts

• universal algebra

References

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

• Samuel Eilenberg, G. Max Kelly, Closed Categories, pp. 421-562 in: S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) [doi:10.1007/978-3-642-99902-4]

• Francis Borceux, Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994)

Discussion more focused on actual universal algebra:

• R. F. C. Walters, A categorical approach to universal algebra, Ph.D. Thesis (1970) [anu:1885/133321]

• Martin Hyland, John Power, The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads, Electronic Notes in Theor. Comp. Sci. 172 (2007) 437-458 [doi:10.1016/j.entcs.2007.02.019, preprint]

Exposition of basics of monoidal categories and categorical algebra:

• geometry of physics – categories and toposes, Section 2: Basic notions of categorical algebra

See also

• Wikipedia, Categorical algebra