re: Query (Q-algebras)

[Vaughan Pratt <pratt@CS.Stanford.EDU>]

The operations of Q-alg are the functions between the powers of the set Z,
forming the (Lawvere) theory T of Q-algebras, which are product-preserving
functors from T to Set.  (So if Z is 3 then there are 27 = 3^3 "Boolean"
operations in place of the familiar 4 = 2^2.)  With all powers Q-algebras are
equivalent to CABAs, with only finite powers and finite-product-preserving
functors they are equivalent to Boolean algebras.

A natural next question would be, what is obtained when T is taken to be
Set itself, in each of the cases when the functors T->Set are required to
preserve all limits, and just the discrete ones?

Vaughan Pratt

>From: Oswald Wyler <owyler@suscom-maine.net>
>For every set Z, there is a self-adjoint contravariant functor Q=ENS(--,Z),
>with unit/counit h:Id-->Q^op Q given by (h_X)(x)(f)=f(x).  Let Q-alg denote
>the category of algebras for the monad induced by this self-adjunction.
>If Z is not empty or a singleton, then the comparison functor ENS^op-->Q-alg
>is an equivalence by results of M. Sobral.  If Z has two members, then Q-alg
>is isomorphic to CaBool, the category of complete atomic Boolean algebras.
>What is known about Q-alg if Z has more than two members (beyond the fact
>that Q-alg and CaBool are equivalent)?