Thread View: sci.math.research
2 messages
2 total messages
Started by Allan Adler
Fri, 25 Dec 1992 06:58
points
Author: Allan Adler
Date: Fri, 25 Dec 1992 06:58
Date: Fri, 25 Dec 1992 06:58
27 lines
1871 bytes
1871 bytes
Let X be a scheme. There are various toposes associated to X, more than I can name here. These include the Zariski topos, the etale topos, the flat topos, the fppf topos, the fpqc topos, the crystalline topos, etc. There is a notion of a point of a topos. A point of a topos T is a morphism of toposes p:Ens-->T, where Ens is the topos of sets. I think that in the case of the Zariski topos, one recovers X as the set of points of the Zariski topos, as least if X is Noetherian and separated, but I could easily be mistaken. At any rate, I would be interested in knowing what are the points of the various toposes associated to X. Is it known in all these cases? I know that in Artin's book Algebraic Spaces, he considers a scheme over an algebraically closed field and at each point x of the scheme, considers the ring of algebraic functions defined at x, while in SGA 4.5 one talks about strictly henselian local rings containing the local rings of X (if I remember correctly). So maybe these are the points of the etale topos, the rings of algebraic functions being a sufficient family of points. Thus, for the Zariski and etale toposes, I have a fraction of an idea, possibly wrong. For the other toposes associated to X, I have no idea. Allan Adler ara@altdorf.ai.mit.edu
Re: points
Author: Alejandro
Date: Tue, 29 Dec 1992 13:09
Date: Tue, 29 Dec 1992 13:09
23 lines
1785 bytes
1785 bytes
I had one or two questions to add to Adler ones, perhaps some naughty cathegorist there can answer easily. First, is the category of conmutative C*-algebras a topos? Being as it is the dual (or it was the antiequivalente?) of the category of compact topological spaces, it would be a topos or sort of, am I correct? Second, what about the category of all C* algebras, this is, including the no conmmutative algebras? Following Connes, you can use ANY C* algebra to make diferential geometry. And you have "points" and "arcs" (R-->M or dually C(M)-->R). This would have a categorial counterpart. Third, which is the difference from the categorial point of view of the two categories? This would be: Which is the correct categorial formulation of conmuttativity? But I m not sure if there are more diffs. I dont know if categorists are or not in the side of Connes but I think they would have something to say about all this stuff of non conmutative geometry, manifolds etc... Alejandro Rivero rivero@cc.unizar.es
Thread Navigation
This is a paginated view of messages in the thread with full content displayed inline.
Messages are displayed in chronological order, with the original post highlighted in green.
Use pagination controls to navigate through all messages in large threads.
Back to All Threads