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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]