### Klein 2-Geometry IX

#### Posted by David Corfield

Seeing Jim Dolan expose that notion of types, predicates and axioms in his lecture, I was reminded of the introduction John gave me to it in Minneapolis. While there, we tried to see what we could make of the idea that a process of categorification moves us up from propositional to predicate to modal logic, (an idea of Jim’s?). What we arrived at was a multi-agent form of S5.

As in the lecture Jim’s example of an axiomatic theory is Euclidean geometry, this led me to a ramshackle series of night thoughts, which is all I’m fit to record at the moment.

So, models of a theory axiomatised in predicate logic assign sets to types, maps from their types to truth values are assigned to typed predicates, and truth values to sentences. The models then form a groupoid. The idea of categorifying predicate logic to modal logic allows metatypes, being assigned groupoids. So,

1) Can we see a degenerate ‘propositional’ 0-geometry?

And,

2) Did we see any sign of something ‘modal’ in our 2-geometry forays?

Maybe in the example I give at the end of this post, doubled-up Euclidean geometry, we might say a single line possibly passes through a point if it passes through it or its twin. While we might say that line-twins necessarily pass through a point, if one of them does.

But doesn’t that doubled-up geometry resemble Connes’s two-sheeted spacetime? And why should this be surprising if many noncommutative algebras arise as the convolution algebra over groupoids? After all, if we’re looking for 2-groups to be symmetries of something, a groupoid seems like a good bet.

If a spectrum is best thought of as a groupoid, what is the spectrum of the convolution algebra on a groupoid?

On a different note, John in a taverna in Delphi suggested thinking about finite 2-groups with a bit of twist to them. So what are the smallest finite groups, $G$ and abelian $A$, such that $H^3(G, A)$ is nontrivial?

## Re: Klein 2-Geometry IX

I have some basic questions about modal logic, which I would like to learn more about. Could you (David) or someone else explain some things to me?

I’m looking for appropriate categorical semantics of the modal operator denoted by a box (“it is necessarily true that…”), something I can really sink my teeth into. Looking at the wikipedia article you linked under S5, it looks like box should be interpreted as a product-preserving, maybe even a left exact comonad, maybe satisfying some additional properties. Axiom T would correspond to a counit, axiom S4 would correspond to a comultiplication, and axiom K to product-preservation.

For example, given a topology on a set $X$, there is an operator

$int: 2^X \to 2^X$

which assigns to a subset of $X$ its interior, and this is a meet-preserving comonad. Would that be a reasonable topological semantics of box, or (thinking of elements of $2^X$ as ‘propositions’) an example of a necessity-modality on a propositional theory? (This reminds me of some things Steve Vickers said in the first few pages of his book Topology via Logic.)

Or, given a category $C$ whose set of objects is $X$, there is a left exact comonad $G$ acting on (what Lawvere sometimes calls) the category of attributes of type $X$, $Set/X$. Here $G$ is defined by the fiberwise formula

$[G(p: Y \to X)]_x = \prod_{f: x \to x'} p^{-1}(x')$

where $x, x'$ belong to $X = Ob(C)$ and $f: x \to x'$ belongs to $Mor(C)$. The category of coalgebras is the topos $Set^C$ (assuming I didn’t get the variance screwed up). Is there some appropriate interpretation of this comonad as a modal box operator?

Is any of this connected with what you and John were discussing?