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/ ]
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 there already a name in the literature for the special instance of duoidal category in which one of the monoidal structures is cartesian? In
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 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/ ]
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/ ]
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/ ]
participants (4)
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David Yetter -
Fred E.J. Linton -
Robert Pare -
Ross Street