Re: Terminology for point-free topology?

[Steven Vickers <s.j.vickers.1@bham.ac.uk>]

Dear Ieke,

Thanks for mentioning that. It's a beautiful paper, both in its results and  in its presentation, and one I still return to.

Another place where I think you were even more explicit was in "The classifying topos of a continuous groupoid I" (1988), where you said -

"... in presenting many arguments concerning generalized, "pointless" spaces, I have tried to convey the idea that by using change-of-base-techniques and exploiting the internal logic of a Grothendieck topos, point-set arguments are perfectly suitable for dealing with pointless spaces (at least as long as one stays within the 'stable' part of the theory)."

(Would you still say that "pointless" and "point-set" are the right phrases  there? I'm proposing "point-free" and "pointwise".)

On the other hand, in your book with Mac Lane, those ideas seemed to go into hiding. In fact I explicitly wrote "Locales and toposes as spaces" as a guide to reading the points back into the book.

My first understanding of these pointwise techniques came in the 1990's, as  I developed the exposition of "Topical categories of domains". That was before I knew those papers of yours, but I felt right from the start that I was merely unveiling techniques already known to the experts - though I hope  you'll agree I've been more explicit about them and particularly the nature and role of geometricity.

I still don't know as much as I would like about the origin and history of those techniques. It would certainly improve my arXiv notes if I could say more.

Might they even have roots in Grothendieck? I once saw a comment by Colin McLarty to the effect that (modulo misrepresentation by me) Grothendieck was  aware of two different lines of reasoning with toposes: by manipulating sites concretely, or by using colimits and finite limits under the rules corresponding to Giraud's theorem. I imagine that as being something like the distinction between pointless and pointwise.

Best wishes,

Steve.



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Hi Steve,

A very early illustration of the strategy of using points in pointless topology is in my paper with Wraith (published 1986). I just looked at it again, and the strategy is explicitly stated in the introduction :

"the strategy is to use adequate extensions of the base topos available from general topos theory, which enable one to follow classical arguments about points of separable metric spaces rather closely. Although both approaches are equivalent, we will follow the second one, because it shows more clearly the interplay between general topos theory and arguments (somewhat similar to those) from topology"

We used it to prove an actual theorem. Of course I used this strategy much more often, e.g. in my two 1990 papers with Joyal.

Ieke


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