### A Question or Two

#### Posted by David Corfield

Points of a set, $X$, correspond to certain maps from the Boolean algebra of subsets, $P(X)$, to $2$, namely those corresponding to prime ideals of the algebra.

Points of a space, $Y$, correspond to certain functors from the topos of locally constant sheaves to Set, via evaluation at a point again. Is there a way to construe this by analogy to the prime ideal story? Is there a ‘spectrum’ around?

How does one characterise the fibre functors to Set which correspond to points? Is there something ‘ideal’ going on?

Cartier’s Mad Day’s Work paper seems to suggest there is such a story going on here. In the same paragraph (p. 404) as the description of the fundamental group as the automorphism group of a fibre functor, he speaks of the Galois group of a field extension in terms of the fields’ spectra.

Posted at February 25, 2008 11:43 AM UTC
## Re: A Question or Two

Here are some quick thoughts on this.

First, a nitpick: if you start with a set $X$, then points $x \in X$ do of course give prime ideals of $P X$ (kernels of Boolean ring maps $eval_x: P X \to \mathbf{2}$). But if $X$ is infinite, then by the axiom of choice, there are other prime ideals too: namely, maximal ideals containing the prime ideal consisting of finite subsets of $X$. These are in bijection with nonprincipal ultrafilters on $X$, which can be considered “ideal points at infinity” that are adjoined to $X$ to form its Stone-Cech compactification.

But we can get $X$ precisely if we change the language, to include arbitrary joins instead of just finite ones. Then an “ideal” $I$ of $P X$ would be defined to be closed under arbitrary joins (and would be a submodule of $P X$ under the meet operation); the prime ideals in that case do correspond to points of $X$. In other words, work in the category of (Boolean) locales, not just (Boolean) rings.

Actually, this is not just a nitpick; I think the change in point of view is important. The idea is that the concept of Grothendieck topos is a categorification of the concept of locale. The analogue of locale map (one which preserves arbitrary joins and finite meets) would be a functor which preserves arbitrary colimits and finite limits, i.e., the left adjoint part of a geometric morphism.

So, for example, suppose we start with a topological space $X$. Under a fairly mild assumption (that $X$ is sober), those functors

$Sheaves(X) \to Set$

which preserve colimits and finite limits correspond exactly to points $x$ of $X$; the corresponding “evaluation” functor $eval_x$ sends a sheaf to its stalk at $x$ (or its fiber, restricting to locally constant sheaves as you mention).

So a first answer to how to characterize these fiber functors is to say they’re the cocontinuous finitely continuous ones to $Set$.

The analogue to ‘ideal’ is less clear-cut. In ring theory, the notion of ideal is a simplification of an appropriate notion of congruence relation, a simplification which is possible because of the presence of additive inverses (i.e., we say $a$ is congruent to $b$ modulo $I$ if $a - b \in I$). Since such congruence relations are kernel pairs of ring maps $f: A \to B$, one could categorify that by taking the kernel pair = weak pullback of a left exact left adjoint $f: E \to F$ against itself:

$E \times_F E \stackrel{\to}{\to} E.$

This kind of thing is studied by the topos theorists, especially in conjunction with descent theory, starting with the Joyal-Tierney monograph An Extension of the Galois Theory of Grothendieck. (I would have to reread that monograph to remind myself of the precise way in which all this is connected with Grothendieck’s Galois theory, and anyway the explanation wouldn’t fit in the margin here.)

In a rather different direction, Steve Awodey in his thesis pursued the following analogy: just as a commutative ring can be reconstituted as the ring of global sections of a sheaf of local rings on its space of prime ideals, so in the 2-category of elementary toposes and

logicalmorphisms, a small topos $E$ could be reconstituted by taking global sections of a sheaf [or actually stack] of [“hyper”]local toposes over a certain site. The first such site is relatively tautological: it is the category $E$ itself equipped with the finite covering topology. The second is more sophisticated: the site is a certaintopological space! Awodey goes on to give some interesting applications to higher-order logic, but as I say this direction would seem to be a little different to the one you want to go in.