* Re: Half cartesian duoical categories
@ 2016-10-07 19:52 Fred E.J. Linton
0 siblings, 0 replies; 4+ messages in thread
From: Fred E.J. Linton @ 2016-10-07 19:52 UTC (permalink / raw)
To: David Yetter, categories
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/ ]
^ permalink raw reply [flat|nested] 4+ messages in thread
* Re: Half cartesian duoical categories
2016-10-06 19:48 David Yetter
2016-10-08 2:57 ` Ross Street
@ 2016-10-08 11:31 ` Robert Pare
1 sibling, 0 replies; 4+ messages in thread
From: Robert Pare @ 2016-10-08 11:31 UTC (permalink / raw)
To: David Yetter; +Cc: categories, Marco Grandis
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/ ]
^ permalink raw reply [flat|nested] 4+ messages in thread
* Re: Half cartesian duoical categories
2016-10-06 19:48 David Yetter
@ 2016-10-08 2:57 ` Ross Street
2016-10-08 11:31 ` Robert Pare
1 sibling, 0 replies; 4+ messages in thread
From: Ross Street @ 2016-10-08 2:57 UTC (permalink / raw)
To: David Yetter; +Cc: categories@mta.ca list
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 4+ messages in thread
* Half cartesian duoical categories
@ 2016-10-06 19:48 David Yetter
2016-10-08 2:57 ` Ross Street
2016-10-08 11:31 ` Robert Pare
0 siblings, 2 replies; 4+ messages in thread
From: David Yetter @ 2016-10-06 19:48 UTC (permalink / raw)
To: 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/ ]
^ permalink raw reply [flat|nested] 4+ messages in thread
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