Rather than give an incorrect or incomplete answer to this question I throw it open to the categories list. I wonder if Max Kelly or Brian Day might have done something along these lines, but I don't know. Michael On Thu, 4 Apr 2013, Patrik Eklund wrote:
Dear Michael,
We were in touch with you some years ago on formal term constructions in term monads, and also as related to distributive laws and monads over various categories. Extensions to many-sortedness where then in focus. Thanks again for those discussions.
We have proceeded in these matters, and now also been communicating with Ulrich H?hle, who has extended these term constructions to work more generally over monoidal (and biclosed) categories. The underlying idea, kind of, is to view the tensor as the 'tensoring of sorts', so that, intuitively, an operator omega : s_1 x ... x s_n -> s is more like omega : s_1 tensor ... tensor s_n -> s. It also keeps the object S of sorts distinct from the actual sorts being elements of the Hom(1,S) using the monoidal category things. This view was inspired by Benabou's 'scheme' notation of signatures.
Ulrich's constructions work suprisingly well and are very natural, so we were wondering about appearence of these results in the literature.
Therefore the following question:
Did anyone ever publish anything on term monad constructions over monoidal categories?
If you have any, even partial answer to this question, we would be much obliged.
We really underline 'construction', and clearly 'term monad'. Similar to conventional term monads (e.g. Manes 1976), formal constructions are very rare, even if we look into Manes' book and other literature, and proceed from there. This was in fact our motivation to work with term functor constructions, and also in the many-sorted setting. Some first motivations appeared when we wanted to compose powerset like monads with the term monad, and then the construction was critical when also constructing the 'swapper' in the distributive laws. I am enclosing a recent paper, where Ulrich helped us on that term constructions.
Anyway, back to 'term monads over monoidal'. There are papers on monads over monoidal categories. Kock did one, "Monads on symmetric monoidal closed categories", in 1971 (can be found under Google), and Wolff continued along similar lines in 1973 (http://link.springer.com/article/10.1007%2FBF01228184?LI=true#). Seal has a recent paper on "Tensors, monads and actions" (http://arxiv.org/abs/1205.0101). In all these approaches, it is more about the 'monoidal monad' (or the strong monad), i.e. than using the tensor as a building block for strengthening the monad conditions,
There are more papers, but they seem all seem to follow the 'monoidal monad' path.
Anders Harvey, Gavin, same question to you obviously. We are much obliged for any partial answer you may have, and pointers to further papers of yours.
Best regards,
Patrik
PS Robert Helgesson in Ume? is finishing a PhD thesis on consequences of these things on institutions (Goguen) and entailment systems (Meseguer), and his defense is planned for May 28 (with Andrzej Tarlecki as opponent). If you eventually want to have a copy of his thesis, please send Robert an e-mail (rah@cs.umu.se).
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