Previously we claimed that if you want to check whether a category “behaves like a category of spaces,” you can try checking whether it’s distributive. The goal of today’s post is to justify the assertion that objects in distributive categories behave like spaces by showing that they have a notion of “connected components.”
For starters, let be a distributive category with terminal object , and let be the coproduct of two copies of . For an object , what does look like? If and is a sufficiently well-behaved topological space, morphisms correspond to subsets of the connected components of , and naturally has have the structure of a Boolean algebra or Boolean ring whose elements can be interpreted as subsets of the connected components of .
It turns out that naturally has the structure of a Boolean algebra or Boolean ring (more invariantly, the structure of a model of the Lawvere theory of Boolean functions) in any distributive category. Hence any distributive category naturally admits a contravariant functor into Boolean rings, or, via Stone duality, a covariant functor into profinite sets / Stone spaces. This is our “connected components” functor. When the object this functor outputs is known as the Pierce spectrum.
This construction can be thought of as trying to do for what the étale fundamental group does for .
In general, in a category with finite products, the set naturally acquires the structure of a model of the Lawvere theory generated by , namely the full subcategory on the finite products . (In our previous post on Lawvere theories we made use of the analogous construction for finite coproducts of .)
How can we calculate this Lawvere theory when in a distributive category? Using distributivity! By induction, it’s not hard to see that is literally the coproduct of copies of in any distributive category. It follows that a morphism is a -tuple of morphisms . There are two distinguished such morphisms, namely the two coproduct inclusions into , and using these two morphisms we can write down for any Boolean function a map , in a way that agrees with products and composition.
(Edit: This section previously contained a mistake which was pointed out by Zhen Lin in the comments.)
More abstractly, if is any distributive category then there is a natural functor given by sending a finite set to the coproduct , and by distributivity this functor preserves finite products in addition to finite coproducts. Above we’re looking at the image of the sets and morphisms between them under this functor, which are the objects and some distinguished morphisms between them.
(Note that distributivity can even be regarded as the crucial ingredient in the proof we outlined previously that Boolean functions can all be generated by constants and the if-then-else ternary operator; the key observation in that proof is that we can distribute , which is what let us express -ary Boolean functions in terms of pairs of -ary Boolean functions using if-then-else.)
It follows that in any distributive category, naturally acquires the structure of a model of the Lawvere theory of Boolean functions (so, according to taste, either a Boolean algebra or a Boolean ring).
Intuitively, a morphism is a way to disconnect into two pieces, namely the pullbacks where the morphism is either of the two coproduct inclusions, although is probably not the coproduct of these two pieces unless we assume the stronger condition that the ambient category is extensive. These can in turn be thought of as subsets of the connected components of , and the Boolean algebra / ring structure then comes from the usual logical operations on such subsets, e.g. intersection and union.
The example of affine schemes
The example of affine schemes is worth working through in detail. First, the terminal object in affine schemes is , and so is . It’s enlightening to rewrite this using the isomorphism
which reveals that is the free commutative ring on an idempotent, and hence that morphisms , for a commutative ring, are naturally in bijection with idempotents in . Idempotents are in turn in bijection with direct product decompositions
and so morphisms of affine schemes really do correspond to ways to disconnect into pieces . Abstractly, this reflects the fact that is extensive, and not only distributive.
The abstract discussion above implies that the set of idempotents in canonically acquires the structure of a Boolean ring, which we’ll denote . The multiplication in is just the usual multiplication on idempotents, but the addition is the following modified addition : if are idempotents, then
Note that the third term disappears if has characteristic . Geometrically we can think of as indicator functions of unions of connected components of ; then the RHS describes the operation that must be performed on indicator functions, regarded as taking values in , to get XOR.
Altogether, we find that taking idempotents gives a functor from commutative rings to Boolean rings. (Curiously, it is not the right adjoint to the inclusion of Boolean rings into commutative rings, although it does preserve limits.) Taking opposite categories, we get a functor from affine schemes to profinite sets called the Pierce spectrum functor, which we’ll denote
consists off a point iff is connected, meaning it has exactly two idempotents, and (which are not equal, so the zero ring is not connected). This condition is equivalent to the Zariski spectrum being connected as a topological space, and holds, for example, for any integral domain.
The Pierce spectrum organizes the Zariski spectrum into “connected components” as follows. If is a prime ideal of , then the quotient map induces a map
on Pierce spectra. Since the Pierce spectrum of is a point, we can associate to a unique point in , which intuitively is the connected component to which the point belongs. This construction organizes into a natural map
where on the LHS denotes the prime spectrum as a topological space. (Curiously, this is a map on underlying topological spaces between two ringed spaces which cannot be promoted to a map of ringed spaces, basically because the natural inclusion of the Boolean ring object into the affine line is not a morphism of ring objects.)
The fibers of this map can be given natural affine scheme structures, as follows. An element of the Pierce spectrum can be thought of as a homomorphism of Boolean rings. These can be regarded as generalizations of ultrafilters, which they reduce to in the special case that is a product of copies of (so that the Pierce spectrum is the Stone-Čech compactification ). This occurs whenever is a product of connected rings (e.g. integral domains).
Accordingly, there is a generalization of an ultraproduct construction we can perform here: given , we can quotient by the ideal generated by the elements of (which, recall, are idempotents in ). The result, which we’ll call in deference to ultraproduct notation, is a connected ring, since every idempotent in belongs to either or and hence gets identified with either or in this quotient, and in fact any morphism from to a connected ring must factor through one of these morphisms . Geometrically this says that any morphism from a connected affine scheme to factors through some , so these quotients really do deserve the name “connected components.”
Example. Let . On math.SE, Martin Brandenburg recently asked what one can say about the Zariski spectrum of , and Eric Wofsey gave an excellent answer in terms of looking at the fibers of over its Pierce spectrum exactly as above (which in fact motivated the writing of this post).
In this case is , so the Pierce spectrum is , which can be identified with the space of ultrafilters on , and the fibers are given by ultraproducts. These ultraproducts admit many interesting prime ideals: for example, if is any sequence of primes, then we get a quotient map
to an ultraproduct of fields, which is again a field. So there is a prime ideal for every “nonstandard prime.”