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 .