Re: sketch theory

[Steve Lack <s.lack@uws.edu.au>]

On 25/05/09 3:03 PM, "John Baez" <john.c.baez@gmail.com> wrote:

> Steve Lack writes:
>
> You ask about a sketch for cartesian closed categories. Have a look at
>> at the paper "A presentation of topoi as algebraic relative to categories
>> or
>> graphs (Dubuc-Kelly, J. Alg. 81: 420-433, 1983). This describes something
>> even tighter: the category of cartesian closed categories is monadic over
>> the category of graphs.
>>
>
> Thanks!
>
> And thanks to everyone else for their helpful comments.  I'm behind on
> answering my emails.
>
> In this approach, does each pair of objects in a ccc come with a chosen
> product and exponential? Are the morphisms of ccc's are required to preserve
> these on the nose?

Yes, that's right on both counts, but see below.

>
> At first I was a bit shocked to hear of a sketch for ccc's, because the
> internal hom is contravariant in one variable.  But I guess that as long as
> we treat ccc's purely 1-categorically that's no problem.  But then I guess
> we pay the price of "undue strictness".  Right?

As you say, if you work 1-categorically, you are stuck with undue
strictness. And as you imply, there is an impediment to a fully
2-categorical approach because of the contravariance of the internal hom.
But there is a way around this. You work 2-categorically, but not over the
2-category Cat, but over the 2-category of categories, functors, and natural
_isomorphisms_. (Kelly & co call this 2-category Cat_g, with g presumably
standing for groupoidal, since this is not just enriched in Cat but in
groupoids.) Then the internal hom does indeed become a 2-functor
Cat^2_g-->Cat_g.

Having these invertible 2-cells now allows you to consider pseudomorphisms
of algebras, which preserve structure up to isomorphism, thus alleviating
the problem of undue strictness. It doesn't completely solve it - since we
only have invertible 2-cells, we don't have a notion of lax morphism; or,
more precisely, the notion of lax morphism we get is just that of
pseudomorphism. Similarly, some constructions might not give what we hoped
for. For example, the cotensor C^2 in Cat_g is not the category of arrows in
C, it's the category of invertible arrows in C.

All the best,

Steve.





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