The n-Category Café

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 logical morphisms, 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 certain topological 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.

The general question is a bit too abstract for my level of competence, but there are much more restrictive situations in which the issues are very deep. The one I’m familiar with is when X is a compact curve of higher genus defined over Q with a rational point b. On the category of finite covering spaces of X, b determines a fiber functor F_b that’s `defined over Q’ in a suitable sense. There is a conjecture that proposes that *all* fiber functors that are defined over Q should come from rational points. This is a version of Grothendieck’s section conjecture.

Some details are contained in my colloquium talk at Leeds

MK