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).


  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?

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.