Re: Small is beautiful

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear Bob,

This reminds me of a distinction that arises topologically. In a
discrete space, equality is "OK" in the the sense that the diagonal is
an open subspace of the square. This works fine also for point-free
spaces, by a result in the Joyal--Tierney monograph. (A space X is
discrete iff all finite diagonals X -> X^n are open maps.)

That suggests working more generally with (point-free) topological
categories: the collections of objects and morphisms are spaces. Then
the ones with object equality OK are the small ones, where the spaces of
objects and morphisms are discrete, i.e. sets.

At first sight this doesn't help us with large categories. But actually
we go a long way if we generalize spaces a la Grothendieck. For example,
the "topologized version of the class of sets" is then the object
classifier S[U], a Grothendieck topos whose points are sets. That may
look like a clumsy way of replacing something unbeautiful (the large
category of sets) by something even worse. But in fact S[U] can be
presented in a way that doesn't presuppose knowledge of all of Set, by
using a site on a small category of finite sets, whose objects are the
natural numbers and whose morphisms correspond to functions between the
finite cardinals.

Many large categories, including categories of structures such as Group,
Ring etc., can be replaced in this way by topical categories, whose
collections of objects and morphisms are toposes and whose domain,
codomain etc. functors are geometric morphisms. Topical functors, again,
are made from geometric morphisms, which imposes continuity conditions
on the functors (e.g. preservation of filtered colimits, analogous to
Scott continuity).

In effect this revises the notion of "class", replacing formulae in set
theory by theories in geometric logic.

Example: In the topical category of groups, the (generalized) space of
objects is the group classifier S[Gp] while the space of morphisms is
S[GpHom], the classifier for pairs of groups with homomorphism between
them. Similarly for rings we have S[Rg] and S[RgHom]. The group ring
construction is then given by geometric morphisms S[Gp] -> S[Rg] and
S[GpHom] -> S[RgHom], satisfying the functoriality conditions.
(Actually, in this example the morphism parts are given canonically once
we have S[Gp] -> S[Rg].)

Best wishes,

Steve Vickers.

Robert Pare wrote:
> ...
> Well, after these ramblings, perhaps my message is lost. So here it is:
> Small categories -> equality of objects okay
> Large categories -> equality of objects not okay
> Small is beautiful, not evil.
>
> Bob


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