Re: Alternative closed structure on Cat

[selinger@mathstat.dal.ca (Peter Selinger)]

Dear Ondrej,

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)

-- Peter

Ondrej Rypacek wrote:
>
> Dear All
>
> Is there a name for and what is known about the tensor product of =
> ordinary categories which looks like the underlying 1-category of Gray's =
> (Gray) tensor product?=20
> Explicitly, roughly:=20
> - objects of C \otimes D are pairs (c,d) , c object of C , d an object =
> of D
> - arrows alternating lists of arrows from C and D, i.e. they are =
> generated by=20
>  	(f,d) : (c,d) -> (c',d) for f : c -> c',
>  	(c,g) : (c,d) -> (c,d') for g : d -> d'
>
> and modulo the equations:   (f',d) . (f, d) =3D (f'f, d), (c,g') . (c,g) =
> =3D (c,g'g), and identities, left and right unit laws and associativity =
> in each component separately.
> =09
>
> And is the category of categories with respect to this tensor closed ?=20=
>
>
>
> Thank you!
> Ondrej
>
>
>
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>



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