Re: generalised cartesian multicategories

[Michael Shulman <shulman@sandiego.edu>]

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


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