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I began writing the article club, with more to come.
Thanks! I added some interspersed remarks about how clubs can be regarded as generalized operads, and asked a question.
Thanks for those additions, Mike. I responded to your question, and will probably export my answer to the text when I get the chance.
I added a short section to club about the general notion of “club over a cartesian monad”, with a reference. (I don’t have at hand the correct references for the other versions; maybe some else can add them.)
This is in preparation for hopefully generalizing it to cartesian monads in 2-categories other than $Cat$, including Garner’s double clubs. Has anyone seen other such generalizations of the notion of club?
I did some work on club, mentioning the finite sets, many-objects, and over-Cat versions, and relating them all to polynomial monads. I am still rather puzzled by the mixed-variance case. I moved the parts of the query-box into the text that were not already there, except for a question and answer that I don’t fully understand. If mixed-variance clubs are the generalized operads for some cartesian monad, what is that monad? It can’t be polynomial. What are its algebras?
Re: #5, now I think maybe its algebras are strictly compact closed categories (i.e. symmetric strict monoidal categories equipped with a strictly involutive choice of duals) that are also enriched over pointed sets and such that all traces are the zero morphism. I can imagine it being cartesian.
Is this version of the mixed-variance case, with the shape category enriched over pointed sets, in one of Kelly’s papers? I can’t find it right now.
Beats me. If I had to guess, I’d guess the social group, but it might be something entirely different.
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