Re: Half cartesian duoical categories

["Fred E.J. Linton" <fejlinton@usa.net>]

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
> 
> 


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