Re: generalised cartesian multicategories

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

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


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