Notions of Lawvere theory

Details Notions of Lawvere theory Notions of Lawvere theory

2008-10-15

arxiv.org/abs/0810.2578v2

Applied Categorical Structures 19(1):363-391, 2011

Mathematics

Category Theory

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

Scientific paper

10.1007/s10485-009-9215-2

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.

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