Dears Fred, Steve and Andree Thank you for your mails. I shall answer you by quoting extracts of your mails and answering as well as I can to these extracts. Steve

I think I understood that, but my point was that if the structure functor for a category of topological spaces was to have codomain Sets, given by space |-> set of points, then these topological examples could be misleading. They are obviously different in nature from the algebraic examples (groups, lattices, etc.), and the idea that they at least have faithfulness in common is a foundational accident.

To define structure functors I need not only faithfulness but also transportability. And to distinguish from structures of algebraic kind like groups; rings, but many others, I add the notion of algebraic structure, which boils down to: p reflects the isos. This immediately eliminates the forgetful functor Space --> Set where Space is the category of topological spaces. It eliminates others, e.g. the forgetful functor Preord --> Set where Preord is the category of preordered sets But plenty of others. Steve again,:

I should check that I understand your reference to toposes correctly, with functors to CAT. Do you just mean the topos as category, in other words - if the topos is generalized space - the category of sheaves? That is actually a point-free gadget. The correct analogy for topological spaces would then be that the structure functor has codomain Poset^op, with space |-> lattice of opens. That is indeed faithful, but it is not the kind of "topological category" discussed in Joy of Cat. In fact it is very much of the algebraic nature You seem to consider that point free notions are an absolute must. First I never mentioned points in my definitions. In the EXAMPLES they might occur. So what? I am at lest as convinced as you are that Topos Theory has been a great achievement in the history of mathematics. But I don't think they are the BEGINNING of of mathematics, i.e. that all that existed before should be either forgotten or at least re-written in topos theoretic versions, nor that it is the END of mathematical concepts i.e. that every part of future mathematics should be written in this framework.

That is why I think one should be suspicious of "commonly agreed" notions of structure functor. They may be trying to encompass misleading examples. What is common between a group, a rational number, and the space of real numbers? They are all objects of vert simple categories, namely: The category of groups; the (pre)ordered set of rational numbers and

Steve again the category of locally compact Hausdorff spaces. Thus Category theory encompasses wildly different notions yet we don't think it is misleading Steve and Fred OK, both of you don't believe in structures, but you do believe in functors. And you know that some functors p are more interesting than arbitrary ones and that one can prove more about them. Let me give some examples: p is faithful p is full and faithful p is flat p has a left adjoint p is a fibration p has a left adjoint which preserves finite limits Suppose, without mentioning the notion of structure I came and told you that the functors p satisfying the following two conditions are interesting and should be studied (i) p is faithful (ii) The restriction of p to isos is a discrete fibration. You'd have thought I have become senile (and maybe I have!) If I had added that, among these functors, those which reflect isos are much better; you'd have called an ambulance for me. In my next mail I shall give a few striking properties of these functors, I know many, many more, and I ask you yo humor me and let me call them structures before you call the ambulance. Best regards, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]