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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?
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