Dear Mike, Thanks a lot for your answer and hint, I'll try to figure this out. The answer to your question
Can you say anything about what it means for "cartesian multicategories" to "make sense" for a monad T?
is: not yet in general. But i can say what it'd like it to mean for my particular monad T = fm fc: for any graph G, consider the span fm(G) -|→ fm(G) defined by ∑ₘ Gᵐ ← ∑_{m,n} mⁿ ⋅ Gᵐ → ∑ₙ Gⁿ (m,e) ↤ (m,n,f,e) ↦ (n, e ∘ f) (both on edges and vertices). If i'm correct, this forms a monad in Span(Gph), by composing underlying maps (f here), say M. Cartesian structure on a T-multicategory E : TG -|→ G consists of an action E ∘ M → E satisfying some axioms to be made precise, e.g., (E M M → E M → E) = (E M M → E M → E), (E → E M → E) = id_E (E E M → E M → E) = (E E M → E E → E) (maybe more?). Concretely, the domain of a morphism in such a T-multicategory is a finite sequence of paths in the underlying graph G, i.e., (ignoring the case of empty paths) tuples of tuples ((e¹₁,…,e¹ₙ₁), …, (eᵖ₁,…,eᵖₙₚ)), where target(eⁱⱼ) = source(eⁱ_{j+1}) (but not, e.g., target(eⁱₙᵢ) = source(e^{i+1}₁) in general). Does that make any sense? Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]