On 9 Jan 2010, at 03:29, Joyal, André wrote:
Dear All
Many people seem to distrust the equality relation between the objects of a (large) category. Is this a philosophical conundrum or a mathematical problem?
Can we define a notion of (large) category without supposing that its (large) set of objects has a diagonal
Dear Andre, Can I explore this with regard to topologies? Suppose we compare FinSet with Set, defining FinSet very small with N for its object space. The object diagonal N -> NxN is an open inclusion. Now look at Set. The natural topology on the class of sets, as the Ind-completion of FinSet, is the one whose sheaves are given by the object classifier. Thus continuous maps from it are functorial and preserve filtered colimits (the categorical analogue of Scott continuity). (This introduces a confusing issue. Functors from the category Set are already determined by their object maps. But it is a special category in which the morphism space is the comma object got from two copies of the identity map on the object space - we are using the fact that the category of spaces is actually a 2-category, using the specialization morphisms. FinSet was certainly not of this kind.) Now the object diagonal is not even an inclusion, since it is not full. I would speculate, by analogy with what I know for ideal completions, that it is essential but not locally connected. I don't really know what to make of this, but it does seem that there are topological distinctions to be made between the two categories based on object equality. Now let me wonder about classifying toposes. I love using them (as I did above), but they always seem slightly fuzzy because they are really defined only up to equivalence. So I certainly would distrust the object equality. I think the discussion becomes slightly sharper in terms of arithmetic universes (as mentioned by Paul Taylor) instead of toposes. Given a base AU A0, there are two obvious places to look for an object classifier (representing the class of sets) a la Grothendieck topos theory. First, there is A0[U], the AU freely generated over A0 by an object U. This can be constructed by universal algebra, and then is a classifying A0-AU (for the theory with one sort and no predicates, functions or axioms) characterized up to isomorphism with respect to strict A0-AU homomorphisms. However, its object equality depends rather delicately on the precise structure used to characterized AUs. For example, I suspect it will differ according as AUs are taken to have canonical pullbacks or canonical binary products and equalizers. On the other hand, sheaf theory would suggest using the category Presh (FinSet^op) of internal diagrams over the internal FinSet in A0. I conjecture that this (is an AU and) is equivalent but not isomorphic to A0[U]. Hence there are issues of object equality when one compares them. (Milly Maietti and I are looking at "subspaces" in this AU setting, and again are having to be rather careful about equality and the distinction between strict and non-strict AU homomorphisms.) A further issue, seen in AUs but not toposes, is that A0[U] can be internalized in A0, as the internal AU in A0 freely generated by one object. The comparison with the external A0[U] will presumably raise truth v. proof issues similar to those that you have already investigated for the initial AU and Goedel's Theorem. To summarize: (1) Granted the existence of the object diagonals, they may still have different topological characteristics for different kinds of categories. (2) A universal characterization (such as for classifying toposes) that is only up to equivalence will make it difficult to rely on object equality. (3) In arithmetic universes we can perhaps (allowing for the fact that the theory is not fully developed yet) see situations where the object equality definitely exists, but is sensitive to inessential differences. Best regards, Steve Vickers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]