A Lawvere theory is a categorical tool for reasoning about algebraic theories.

Multi-sorted algebraic theory

Enriched Lawvere theory

Lawvere 2-theory

William Lawvere?, Functorial Semantics of Algebraic Theories? , Ph.D. thesis Columbia University (1963). Published with an author’s comment and a supplement in: Reprints in Theory and Applications of Categories 5 (2004) pp 1–121. (abstract)

Julia E. Bergner?, Rigidification of algebras over multi-sorted theories, Algebraic & Geometric Topology 6, pages 1925-1955, 2006. arXiv:0508152

Stephen Lack, Jiri Rosicky, Notions of Lawvere theory, Applied Categorical Structures 19(1):363-391, 2011. arXiv:0810.2578

Daniel Cicala, nCafe post, A Discussion on Notions of Lawvere Theories, 2017.

nLab, Lawvere theory