Subject: categories: RE: Why binary products are ordered

From: S Vickers <[email-masked]>
Date: Thu, 01 Feb 2001 11:10:06 +0000

[David Benson asked me:
>This looks extremely useful...  Where might I read more about this
>formulation of limits?
>
>> The product of f: X -> M is then a universal solution to the problem of
>> finding
>> 
>>    g: YxM -> X  over M
>> 
>> and this is equivalent to the usual characterization once you have chosen
>> the isomorphism between M and 2.
>
Perhaps readers of the list can suggest sources I'd overlooked.] 

Dear David,

The underlying idea is a very broad one, namely to look for limits of
internal diagrams. You will find these described in Johnstone's "Topos
Theory" and (I think - I haven't got it to hand) Mac Lane and Moerdijk.

Suppose, working in a suitable category S (let's say a topos), you have an
internal category C. Then you can define a notion of internal diagram over
C (C-shaped diagram of S-objects) and they are the objects of an external
category S^C, another topos.

If D is another internal category in S and F: C -> D is a functor, then
there is a functor F^*: S^D -> S^C defined using pullbacks in S and it has
both right and left adjoints. In fact it defines an essential (if I
remember the right word) geometric morphism from S^C to S^D.

Now consider the situation when D is the terminal category, one object and
one morphism. S^D is just S. The the right and left adjoints of F^*
calculate internal limits and colimits of internal diagrams. This is easy
to see, since F^*(X) is just the constant diagram, X everywhere, and a
morphism (natural transformation) between an internal diagram D and F^*(X)
is just a cone or cocone over D from or to X. The adjunctions then express
exactly the universal properties of limits and colimits.

I was describing situations where C was a discrete category, so the
internal limit is an internal product.

I also considered the question of whether the construction of the internal
limit was geometric - preserved under inverse image functors of geometric
morphisms. I have written about this kind of question in my own work, for
instance "Topical Categories of Domains". (See my website,
http://mcs.open.ac.uk/sjv22 .) Geometricity in general rules out the use of
exponentiation in the topos, but exponentiation Y^X is geometric if X is
finite with decidable equality (Johnstone and Wraith, ??"Algebraic theories
in a topos"). My proposal is that the record type construction should be
seen as a solution to the problem of internal products, but that it relies
on finite decidability of the set of field names - hence it's related to
geometricity of the internal product.

In a different direction, note that internal products make sense in
categories other than toposes, even if they don't always exist. For
instance, if f: X -> Y is a continuous map of spaces, then the internal
product, if it exists, is a space of continuous sections of f. I bet that's
explored somewhere in the literature, but I don't know where.

Best regards,

Steve.