# 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?