Subject: categories: Re: Continuity

From: [email-masked] (Steven Vickers)
Date: Tue, 24 Nov 1998 17:42:53 +0000

>Dear categoricians,
>
>I'm wondering about the possibility of speaking of trajectories in a
>category. That is, given a category C, in what sense - if any - can we
>consider a continuous function R->C (where R are the real numbers).
>What extra structure is needed on C to consider it as a topological
>space? what relation with the category structure?
>I guess there's a link with topos theory, but I can't make it really
>clear...
>Subsidiary question: in what case is there a notion of differentiability
>for the function f?
>Thank you for direct answers or pointers to (quite easyly
>understandable) literature.
>Regards,
>----------------------------------
>Matthieu Amiguet, doctorant

The key is indeed via toposes, specifically Grothendieck toposes. As has
always been recognized, these are generalized topological spaces and the
geometric morphisms are generalized continuous maps.

Instead of a category C whose objects are XXX's, you replace it by a topos
whose points are XXX's (i.e. the classifying topos for the theory of
XXX's). For instance, the category of set would be replaced by the
classifying topos for the theory of sets (i.e. what's described in
Johnstone's book and elsewhere as the "object classifier"). I'll write
S[set] for this classifying topos.

This works provided the objects and morphisms of C are the models and
homomorphisms of a geometric theory.

Turning to topological spaces X, if X is sober then its points are also the
models of a geometric theory (to do proper justice to this you need to take
the localic approach to topology) and so there is a corresponding
classifying topos SX which is actually the category of sheaves over X.

The "continuous maps" between toposes are just the geometric morphisms.
They transform points of the source topos to points of the target.

Specific examples:

1. For two toposes SX and SY obtained from topological spaces, the
geometric morphisms from SX to SY really do correspond to the continuous
maps from X to Y.

2. Geometric morphisms from SX to S[set] are the sheaves on X, and these
can indeed be thought of as continuous maps from X to the category of sets
(each sheaf maps points of X to their stalks).

By no means can every category C be considered the category of models of a
geometric theory. One necessary condition (not sufficient) is for C to have
all filtered colimits. For a small category C, one can make a corresponding
topos S^C (the functor category from C to sets) that is an analogue of the
ind completion (which freely adjoins filtered colimits). Its points are the
flat presheaves over C and include (by the Yoneda embedding) a full and
faithful image of C.

A particular link with category structure lies in specialization. Any sober
topological space has the specialization order on its points, and a topos
has analogous specialization morphisms between points - they are just
homomorphisms between models of the geometric theory. For a topos SX
obtained from a space, this "specialization category" structure on the
points is a poset, the original specialization order. Any geometric
morphism is, when considered as a points transformer, functorial with
respect to the specialization morphisms - this generalizes the topological
fact that a continuous map is monotone with respect to the specialization
order.

Though the "generalized space" ideas are not at all new, they are often
hidden expositionally by the idea of topos as generalized universe of sets.
If you look at my paper "Topical Categories of Domains" (to appear in Maths
Structures in Computer Science but at present available through my Web home
page) you will find a careful attempt to discuss toposes directly as
generalized spaces.

The home page is

   http://theory.doc.ic.ac.uk:80/people/Vickers/

There is also a short paper "Strongly Algebraic = SFP (Topically)" that
includes a summary in these purely spatial terms of established results
about points of presheaf toposes.

Regarding differentiability, I too would be interested to hear of answers
or pointers. I guess it would most conveniently be discussed using
Caratheodory's approach. If X is a topological ring, or a ring object in
the category of toposes, then f: X -> X is C^1 iff there is some continuous
phi: XxX -> X such that

   for all x, y. f(y) - f(x) = (y-x).phi(x,y)

(The use of quantified "elements" of X can easily be translated into the
equality of two geometric morphisms from XxX to X.)

The derivative of f is f'(x) = phi(x,x).

Steve Vickers.