Re: generalised cartesian multicategories

Re: generalised cartesian multicategories

[Michael Shulman]

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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generalised cartesian multicategories

[Tom Hirschowitz]

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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Re: generalised cartesian multicategories

[Tom Hirschowitz]

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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Re: generalised cartesian multicategories

[Michael Shulman]

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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