Ok, here's a more precise version of my guess. On Tue, Jul 8, 2014 at 12:06 AM, Tom Hirschowitz <tom.hirschowitz@univ-savoie.fr> wrote:
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.,
Tom clarified by private email that if the definition of M from G is denoted M_G, then in the last paragraph above he means M_{fc(G)}, so that M : TG -|→ TG and hence E ∘ M : TG -|→ G. A monad in a bicategory of spans is, of course, an internal category. I suspect that your construction G |→ M_{fc(G)} can be extended to a monad on the bicategory of internal profunctors in Gph, and that your cartesian T-multicategories are generalized multicategories for this monad (which are "object-discrete" in the sense of my paper with Geoff that I cited in my last email, since their underlying object is a graph G rather than an internal category in graphs). Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]