🚀 go-pugleaf

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