almost bi-monoidal categories

almost bi-monoidal categories

[Ondrej Rypacek]

Dear category theorists

Could anyone kindly help me with the following:
How much is known about categories with

- two monoidal structures
- and a natural transformation (but not isomorphism)  X x Y -> X + Y

I believe this isn't called a bimonoidal category, as we don't have an
iso above (?)


More generally, how about a tricategory with directed interchange law
(X x Y ) + ( W x Z ) -> (X + W) x (Y + Z) ?

Many thanks,
Ondrej


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Re: almost bi-monoidal categories

[Vaughan Pratt]

Not that this answers your question, but keep an eye out for the related
forms

X x (A + Y) --> (X x A) + Y
and
(X + A) x Y --> X + (A x Y)

in case you run into either one.  These are half of the weak
distributivity laws for linear logic studied by Cockett and Seely a
decade or so ago.

Although full-blown category theory didn't exist in the 19th century it
did have its posetal fragment, and the above first appears in C.S.
Peirce "Note B: The Logic of Relatives", 1883, see p. 456 of Vol. 4 of
Kloesel's "Writings of C.S. Peirce" where x and + are respectively
relative product (i.e. composition in Rel) and its De Morgan dual (with
respect to Boolean complement) relative sum.

Peirce describes them as "two formulae that are so constantly used that
hardly anything can be done without them."  (They also hold for x,+ as
logical or Boolean conjunction,disjunction whence "hardly anything"
could well extend to brushing one's teeth etc, though only your
subconscious would know that.)

The posetal case of the internal hom, what Ward and Dilworth called
"residuation" in 1939, goes back even further, namely to De Morgan's
Theorem K in his "On the Syllogism: IV", 1860, first pointed out by
Roger Maddux, see http://boole.stanford.edu/pub/ocbr.pdf (evening LICS
talk I gave 6 weeks after my quintuple bypass).

Vaughan Pratt

On 5/9/2012 4:35 PM, Ondrej Rypacek wrote:
> Could anyone kindly help me with the following:
> How much is known about categories with
>
> - two monoidal structures
> - and a natural transformation (but not isomorphism)  X x Y ->  X + Y
>
> I believe this isn't called a bimonoidal category, as we don't have an
> iso above (?)


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