# Notions of connected components in a finite family fibration

Let $$\Pi_0:\mathsf{FinFam}(\mathsf C)\to \mathsf{FinSet}$$ be the fibration exhibiting the free finite coproduct completion of $$\mathsf C$$. Suppose $$\mathsf C$$ has finite limits so that the extensive $$\mathsf{FinFam}(\mathsf C)$$ is also distributive.

Any distributive category admits a canonical functor $$\mathrm{Hom}(-,\mathbf 2)$$ which lands in the category of Boolean algebras, or - by Stone duality - in the category of profinite spaces.

I don't know much about the topological case, but at least in the case of affine schemes, $$\mathrm{Hom}(-,\mathbf 2)$$ takes an affine scheme to the Boolean algebra of its idempotents. On the other hand, restricting to the category of $$\mathsf{FinFam}(\mathsf{ConnCRing}^\text{op})$$ comprised (I think) of commutative rings with finitely many idempotents, it seems $$\Pi_0$$ sends an affine scheme to its set of minimal idempotents.

There's more structure: suppose $$\mathsf{C}$$ admits a terminal object, which is the same as saying that the terminal functor from $$\mathsf C$$ has a right adjoint (picking the terminal object), then applying the 2-functor $$\mathsf{FinFam}$$ preserves this adjunction giving an adjunction $$\Pi_0\dashv \mathrm{disc}$$. I don't know about the general case of distributive categories, but at least for affine schemes $$\mathrm{Hom}(-,\mathbf 2)$$ has a right adjoint taking a profinite set to the algebra of continuous maps from the profinite set to a given discrete ring.

The unit of $$\Pi_0\dashv \mathrm{disc}$$ takes a "point" to the connected component containing it. In practice $$\mathrm{disc}$$ is often fully faithful, so the counit is an isomorphism (topologically intuitive).

Questions.

1. Does $$\mathrm{Hom}(-,\mathbf 2)$$ of a distributive category admit a right adjoint under some general conditions? What are the unit and counit?
2. How are the structures of $$\Pi_0\dashv \mathrm{disc}$$ and $$\mathrm{Hom}(-,\mathbf 2)$$ (hopefully with a right adjoint) related via $$\mathsf{FinSet}\to\mathsf{Pro}(\mathsf{FinSet})$$?
3. This is basically a subquestion of (2), but given a commutative ring with finitely many idempotents, how to formally reconstruct its Pierce spectrum from its minimal idempotents?