generalised cartesian multicategories

[Tom Hirschowitz <tom.hirschowitz@univ-savoie.fr>]

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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