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
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