ct.category theory – Stone duality for the algebra of Boolean functions such that $f(top,dots,top) = top$, or: What does the presheaf topos on $FinSet_ast$ classify?

It’s well-known that the topos of presheaves on the category $FinSet_{neq emptyset}$ of finite nonempty sets is the classifying topos for boolean algebra objects.

The argument goes like this: Up to idempotent completion, $FinSet_{neq emptyset}$ is equivalent to the category $FinSet_{2^bullet}$ of sets with cardinality a finite power of 2. By Stone duality, we have in turn $FinSet_{2^bullet} simeq FinBool^{op}$, where $FinBool$ is the category of finite boolean algebras, which are necessarily free. Since the theory of boolean algebras is a finitary algebraic theory, finite-product-preserving functors $FinBool^{op} to mathcal E$ are identified with Boolean algebra objects in $mathcal E$, for any category $mathcal E$ with finite products. If $mathcal E = Set$, then any finite-product-preserving functor $FinBool to Set$ is automatically flat; the proof uses the fact that every finitely-presentable Boolean algebra is free. So a geometric morphism $Psh(FinSet_{neq emptyset}) to Set$ is just a Boolean algebra. Then because $Psh(FinSet_{neq emptyset})$ is a prehseaf topos and the theory of Boolean algebras is algebraic, this classifying topos identification extends to all Grothendieck toposes $mathcal E$.


But what happens when we perturb the input a bit? For example, what happens when we consider presheaves on the category $FinSet_ast$ of finite pointed sets? We can, as before, pass to the category $mathcal T = FinSet_{ast,2^bullet}^{op}$ of pointed sets whose cardinality is a finite power of 2. The category $mathcal T$ is a Lawvere theory with half as many operations of each arity as the Lawvere theory for Boolean algebras. Indeed, by identifying the basepoint of a finite pointed set with $top$, we can regard $mathcal T$ as the Lawvere theory of Boolean functions $f$ such that $f(top,dots,top) = top$ — i.e. if all inputs are true, the output must also be true. I believe that $mathcal T$ is generated by the operations $top, wedge, vee, to$, but I’m not sure what a complete set of relations would be.

As a Lawvere theory, $mathcal T$ may be rather messy. But we’re considering something more refined: a finite-product-preserving functor $A: mathcal T to Set$ is not automatically flat. Flatness is equivalent to requiring that $A$ is a filtered colimit of free finitely-generated algebras. In particular, every finitely-generated subalgebra of $A$ is contained in a finitely-generated free algebra. It follows that $A$ will satisfy any universal statement satisfied by all finitely-generated free $mathcal T$-algebras — not just the algebraic ones. I think (but I’m not certain) that the converse is also true — so that $A$ is flat if and only if it satisfies the universal theory of free $mathcal T$-algebras.

Questions:

  1. What is an axiomatization of the algebraic theory of the logical connectives $top, wedge, vee, to$ (i.e. the theory of connectives such that $f(top, dots, top) = top$ for all connectives $f$)?

  2. What is an axiomatization of the universal theory of these logical connectives (i.e.– I think — the theory classified by the presheaf topos $Psh(FinSet_ast)$)?

  3. How do these theories compare to the theory of Boolean algebras — what are some examples of algebras or algebra homomomorphisms which don’t come from Boolean algebras or homomorphisms thereof? Is there a form of Stone duality for these algebras, and how do the corresponding spaces compare to Stone spaces?