Notions of Lawvere theory

Abstract

Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of universal algebra, can be generalized in three ways: replacing Set by another category, working in an enriched setting, and by working with another class of limits than finite products. An important special case involves working with sifted-colimit-preserving monads rather than filtered-colimit-preserving ones.

27 pages. v2 minor changes, final version, to appear in Applied Categorical Structures

References in corpus (2)

• Limits of small functors
• Notes on enriched categories with colimits of some class

Cited by in corpus (16)

• Coend calculus
• Monads and theories
• The Gray tensor product via factorisation
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• Positive fragments of coalgebraic logics
• Distributive laws for Lawvere theories
• Virtual concepts in the theory of accessible categories
• An Equational Metalogic for Monadic Equational Systems
• Two-dimensional regularity and exactness
• Partial groups as symmetric simplicial sets
• Adjoint functor theorems for lax-idempotent pseudomonads
• Towards enriched universal algebra
• More on soundness in the enriched context
• Lawvere theories, finitary monads and Cauchy-completion
• An elementary characterisation of sifted weights
• Categories as models on a suitable algebraic theory