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.