### First-Order Logical Duality

#### Posted by David Corfield

This is the title of the PhD thesis in which Henrik Forssell presents

…an extension of Stone Duality for Boolean Algebras from classical propositional logic to classical first-order logic. The leading idea is, in broad strokes, to take the traditional logical distinction between syntax and semantics and analyze it in terms of the classical mathematical distinction between algebra and geometry, with syntax corresponding to algebra and semantics to geometry.

Extending the duality between Boolean algebras and Stone spaces, Forssell derives a duality between Boolean coherent categories and topological groupoids. A Boolean coherent category corresponding to a first-order theory has as dual the topological groupoid of its models and isomorphisms.

So,

Analogously with the Makkai duality, we consider the groupoid of models and isomorphisms of a theory. Analogously with Stone duality, we use topological structure to equip the models and model isomorphisms of a theory with sufficient structure to recover the theory from them. The result is a first order logical duality which, in comparison to Makkai’s, is more geometrical, in that it uses topology and sheaves on spaces and topological groupoids rather than ultraproducts, and that moreover specializes to the traditional Stone duality. (p. 17)

The dualizing object (did we give up on ambimorphic?) has shifted from $2$ to $Sets$. Next stop $Groupoids$.

## Re: First-Order Logical Duality

So, Jim Dolan has spent many years telling us about the duality between first-order logic and groupoids: a

theoryin first order logic has a groupoid ofmodels, and any groupoid arises as the models of an essentially unique theory. Todd Trimble made this very precise and wrote up a portion here:Most of this concerns the special case of

groups, but his last sentence gives the generalization:From your quote above, it sounds that Makkai has done related work.

So, it would be very nice to 1) let Henrik Forssell know about Todd’s guest posts, 2) understand what Forssell has done.

As for 1): do you know Forssell? Maybe you can send him an email about this blog entry.

As for 2):

I’m really curious what’s the intuitive meaning of a

topologicalgroupoid of models. First, a point of terminology: is this being used to mean a groupoid internal to $Top$ (where we have a topology on the space of objects and on the space of morphisms) or a groupoid enriched in $Top$ (where we only have a topology on each hom-set)? For no particular reason I bet he means the first. In the first case: what does it mean, intuitively, for two models to be close to each other? In either case: what does it mean for two maps between models to be close to each other?Also:

How, if at all, is this stuff related to the Joyal–Tierney theorem showing that Grothendieck topoi are all categories of sheaves on localic groupoids?

Memoirs of the A.M.S.309(1984).‘Localic groupoids’ are a lot like topological groupoids, since the concept of locale is just an improved version of the concept of topological space, with better properties.

A localic groupoid is a groupoid

internal tothe category of locales.It’s a bit hard for me to imagine a result in heavy-duty categorical logic that would work better for topological groupoids than localic groupoids!