Re: Alternative closed structure on Cat

[Ross Street <ross.street@mq.edu.au>]

On 05/07/2012, at 11:38 PM, Peter Selinger wrote:

> Power and Robinson state in [1, Section 2] that the tensor you
> describe is indeed part of a monoidal closed structure: the function
> category has as objects all functors, and as morphisms the (not
> necessarily natural) transformations. Moreover, Power and Robinson
> state that this is the unique other symmetric monoidal closed
> structure on Cat, i.e., there are no others besides this one and the
> "usual" one. I have never seen a proof of this last fact.
> 
> [1] J. Power and E. Robinson. "Premonoidal categories and notions of
> computation." Mathematical Structures in Computer Science 7(5):
> 445-452, 1997. (www.eecs.qmul.ac.uk/~edmundr/pubs/mscs97/premoncat.ps)

Yes, this is what I call the "funny" tensor product on Cat.

Categories enriched in Cat with the funny tensor product are
called "sesquicategories": they are less than 2-categories as they
have whiskering but only ambiguous horizontal composition of 2-cells.

There is a bit of literature on all this. For example, it is mentioned in

Categorical structures, Handbook of Algebra Volume 1 (editor M. Hazewinkel; Elsevier Science, Amsterdam 1996; ISBN 0 444 82212 7) 529-577.

and/or

Higher categories, strings, cubes and simplex equations, Applied Categorical Structures 3 (1995) 29- 77 & 303; MR96b:18009.

As to finding all the symmetric monoidal closed structures on a locally finitely presentable category, 
the object-in-two-categories technique is provided by

F. Foltz, GM. Kelly and C. Lair, Algebraic categories with few monoidal biclosed structures or none, 
J. Pure and Applied Algebra 17 (1980) 171–177.

Perhaps they even give the Cat example.

Best wishes,
Ross

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