# Relation between monads, operads and algebraic theories

I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any reference that treat systematically the relation between such models of theories, where model means a presentation of theory?

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When you say "such models of theories", it's not clear to me whether you mean model in a more model-theoretic sense (which here would be, e.g., "algebra of a theory") or a way of presenting a theory (so an operad would be "such a model of theory"). I guess you mean the latter though. – Todd Trimble Jun 1 '11 at 10:57
You guessed right, by the way I've edited the post, thanks for the correction. – Giorgio Mossa Jun 1 '11 at 13:13

## 3 Answers

All nice recommendations; some more

### Lawvere theories and monads

For the connection between monads and Lawvere theories, I've found this a really nice exposition of the $\mathbf{Set}$ case:

Martin Hyland, John Power - The category theoretic understanding of universal algebra: Lawvere theories and monads - Electronic Notes in Theoretical Computer Science (pdf at M Hyland's website)

and if what you want is something like the ultimate monads <-> Lawvere theories correspondence:

Clemens Berger, Paul-André Melliès, Mark Weber - Monads with arities and their associated theories (arXiv)

Basically all of the (Lawvere and the like) theories <-> (special) monads equivalences can be seen as special cases of their general monad with arities <-> theories with arities equivalence. This paper is a real joy to read, and how all sort of nerve theorems can be viewed as an instance of the equiv between a monad with arities and the corresponding (generalized) algebraic theory is just wonderful!

Also, on the operadic side, their notion of homogeneous theory is related with $T$-operads (with $T$ cartesian and local right adjoint), and yields a nice account of symmetric operads as homogeneous theories with arities the Segal category $\Gamma$.

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Apart from Todd's recommendations, which I'd second, for monads and Lawvere theories there is Nishizawa and Power, Lawvere theories enriched over a general base, JPAA 213, 2009, and the references therein. I'd also recommend Linton's An outline of functorial semantics, LNM 80, 1969 (republished in TAC Reprints). Lots of people have defined 'generalized multicategories' of one sort or another, going back to Burroni in 1971: Cruttwell and Shulman's A unified framework for generalized multicategories, TAC 24(21), 2010, is a useful account.

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Of course, there Tom Leinster's book Higher Operads, Higher Categories, and there's also a lot of stuff on the nLab. Also see Max Kelly's seminal paper (I believe unpublished until recently), On the operads of J.P. May.

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