nLab universal algebra

Idea

Universal algebra (also categorical algebra) is the study of algebraic theories and their models or algebras. Whereas abstract algebra studies groups, rings, modules and so on — that is, models of particular theories — universal algebra is about algebraic or equational theories in general.

Traditionally, the subject studies models of algebraic theories in the category of sets. The category-theoretic approach abstracts the traditional notions, to study models in more general categories. There are several ways of doing this, such as by using monads, Lawvere theories, or type theory.

As with the category-theoretic understanding of many other branches of mathematics, the advantage of doing things this way is not so much the obtaining of new results as the unification of many previously disparate points of view. Examples might include how a Hopf algebra is the same thing as a model in a category of vector spaces of the theory of groups, or how computational side-effects in the theory of programming languages may be understood in terms of free algebras.

Related entries

• algebraic theory

• Lawvere theory

• monad

• operad

• algebra over a Lawvere theory

• algebra over a monad

• algebra over an operad

• Eilenberg-Moore category

• universal algebra in a monoidal category

• algebra

• bar construction

• generalized ring

• Omega-group

• PRO

References

• Fred Linton, §7 in: An outline of functorial semantics, in Seminar on Triples and Categorical Homology Theory, Lecture Notes in Mathematics 80, Springer (1969) 7-52 [doi:10.1007/BFb0083080]

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

• Saunders MacLane, §VI of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971) [doi:10.1007/978-1-4757-4721-8]

• 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]

• Max Kelly and John Power, Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, JPAA 89, 1993.

• Chengming Bai, Li Guo, Jean-Louis Loday, Proceedings of the China-France Summer Institute on Operads and Universal Algebra, World Scientific 2011 (web)

A short introductory/overview note:

• Anthony Voutas, The basic theory of monads and their connection to universal algebra, (2012) [pdf, pdf]

• Andreas Nuyts, Understanding Universal Algebra Using Kleisli-Eilenberg-Moore-Lawvere Diagrams, (2022) [pdf]

Discussion via sheaves of quotient algebras:

• Michael Johnson and Shu-Hao Sun, Remarks on representations of universal algebras by sheaves of quotient algebras, Proceedings of the 1991 Summer Category Theory Meeting, 1992. (pdf)