Re: generalised cartesian multicategories

[Michael Shulman <shulman@sandiego.edu>]

Can you say anything about what it means for "cartesian
multicategories" to "make sense" for a monad T?

There is a more general notion of generalized multicategory which
takes place in a more general bicategory (or, better, a double
category) than T-spans, and which includes cartesian multicategories
as a special case (see
http://tac.mta.ca/tac/volumes/24/21/24-21abs.html for a unified
account, as well as references to a lot of prior work).  I suspect
that your "cartesian T-multicategories" are probably generalized
multicategories in this sense relative to some other monad built out
of T.

Mike

On Fri, Jul 4, 2014 at 7:34 AM, Tom Hirschowitz
<tom.hirschowitz@univ-savoie.fr> wrote:
>
> 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


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