Can you say anything about what it means for "cartesian multicategories" to "make sense" for a monad T? There is a more general notion of generalized multicategory which takes place in a more general bicategory (or, better, a double category) than T-spans, and which includes cartesian multicategories as a special case (see http://tac.mta.ca/tac/volumes/24/21/24-21abs.html for a unified account, as well as references to a lot of prior work). I suspect that your "cartesian T-multicategories" are probably generalized multicategories in this sense relative to some other monad built out of T. Mike On Fri, Jul 4, 2014 at 7:34 AM, Tom Hirschowitz <tom.hirschowitz@univ-savoie.fr> wrote:

Dear all,

Cartesian multicategories are multicategories equipped with `contraction' and `weakening' operations. E.g., contraction associates to any morphism x₁, …, xₙ → y and 1 ≤ i ≤ n such that xⱼ = x_{j+1} for some j a morphism x₁, …, xⱼ, x_{j+2}, … xₙ → y.

On the other hand we have generalised multicategories, which are monads in the bicategory of T-spans, for some cartesian monad T.

I'm currently considering such a monad T for which cartesian multicategories make obvious sense, and wonder whether anyone has worked out a general setting for this. I.e., are there some known conditions on the monad T for cartesian T-multicategories to make sense? Of particular interest would be a setting in which free cartesian T-multcategories exist (over T-graphs).

For those interested, the monad in question is on graphs. It's the composite of

- the `free category' monad fc, and

- the `free monoidal graph' monad fm, mapping any graph s,t : E → T to s*,t* : E* → T*,

made into a monad via the obvious distributive law

fc ∘ fm → fm ∘ fc.

Any hints? Tom

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