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

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

• Francis Borceux, Handbook of Categorical Algebra, III Vols.

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