Half cartesian duoical categories

Half cartesian duoical categories

[David Yetter]

Is there already a name in the literature for the special instance of duoidal category in which one of the monoidal structures is cartesian?  In particular the instance in which if # denotes the non-cartesian monoidal structure and x the cartesian, the lax middle-four interchange transformation has components


(A x B) # (C x D) ------> (A # B) x (C # D) ?


It has come up in my current student's dissertation work.  An existing name  and citations to papers using this specific type of duoidal category would  be much appreciated.


Best Thoughts,

David Yetter

Professor of Mathematics

Kansas State University


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Re: Half cartesian duoical categories

[Ross Street]

Dear David

On 7 Oct 2016, at 6:48 AM, David Yetter <dyetter@ksu.edu<mailto:dyetter@ksu.edu>> wrote:

  In particular the instance in which if # denotes the non-cartesian monoidal structure and x the cartesian, the lax middle-four interchange transformation has components


(A x B) # (C x D) ------> (A # B) x (C # D) ?

You mean, of course,
(A x B) # (C x D) ------> (A # C) x (B # D)

It has come up in my current student's dissertation work.  An existing name   and citations to papers using this specific type of duoidal category would  be much appreciated.

I can't do much better than ``monoidal category with finite products''.
Any such becomes duoidal in this way.

Best wishes,
Ross


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Re: Half cartesian duoical categories

[Robert Pare]

Hi David,

You'll find a lot of information on this sort of duoidal category and variations thereon in our
paper "Intercategories: a framework for three-dimensional category theory" available as #53 at
http://www.mscs.dal.ca/~pare/publications.html
though there is no special name given for it there (as you were asking).

Bob (&Marco)

On 2016-10-06, at 4:48 PM, David Yetter wrote:

> Is there already a name in the literature for the special instance of duoidal category in which one of the monoidal structures is cartesian?  In particular the instance in which if # denotes the non-cartesian monoidal structure and x the cartesian, the lax middle-four interchange transformation has components
> 
> 
> (A x B) # (C x D) ------> (A # B) x (C # D) ?
> 
> 
> It has come up in my current student's dissertation work.  An existing name  and citations to papers using this specific type of duoidal category would  be much appreciated.
> 
> 
> Best Thoughts,
> 
> David Yetter
> 
> Professor of Mathematics
> 
> Kansas State University
> 


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Half cartesian duoical categories

[Fred E.J. Linton]

Fix a duoidal category as in David's scenario, below. Write 1 and I
for the monoidal unit objects for x and #, respectively. If indeed

>  (A x B) # (C x D) ------> (A # B) x (C # D) 

is to hold, then x and # must essentially coincide. Here's why:

1) I = I # I = (1 x I) # (1 x I) = (1 # I) x (1 # I) = 1 x 1 = 1 ; whence

2) A # C = (A x 1) # (C x 1) = (A # 1) x (C # 1) = A x C .

Or perhaps David meant to posit the more usual middle 4 interchange law

: (A x B) # (C x D) ------> (A # C) x (B # D) ?

Wouldn't surprise me. But there I'm no help, sorry. Cheers, -- Fred

---

------ Original Message ------
Received: Fri, 07 Oct 2016 02:50:44 PM EDT
From: David Yetter <dyetter@ksu.edu>
To: "categories@mta.ca" <categories@mta.ca>
Subject: categories: Half cartesian duoical categories

> Is there already a name in the literature for the special instance of
duoidal category in which one of the monoidal structures is cartesian?  In
particular the instance in which if # denotes the non-cartesian monoidal
structure and x the cartesian, the lax middle-four interchange transformation
has components
> 
> 
> (A x B) # (C x D) ------> (A # B) x (C # D) ?
> 
> 
> It has come up in my current student's dissertation work.  An existing name 
and citations to papers using this specific type of duoidal category would  be
much appreciated.
> 
> 
> Best Thoughts,
> 
> David Yetter
> 
> Professor of Mathematics
> 
> Kansas State University
> 
> 


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]