### 2-Galois and 2-Logic

#### Posted by David Corfield

Let’s boldly venture on with our ascent of Mount 2-Logic.

Todd told us about Galois connections via relations

any relation $R \subseteq A \times B$ whatsoever can be used to set up a Galois connection between subsets $A'$ of $A$ and subsets $B'$ of $B$: $\frac{A' \subseteq \{a \in A: \forall_{b: B'} (a, b) \in R\} }{B' \subseteq \{b \in B: \forall_{a: A'} (a, b) \in R\} } iff$

And he also told us that the Galois connection relating theories to symmetry groups works by way of a relation:

In our case, $A$ is the group of permutations $g$ on $X$, $B$ is the set of finitary relations $p$ on $X$, and the relation $R$ is the set of $(g, p)$ such that $p(x_1, \ldots, x_n) = p(g x_1, \ldots, g x_n)$ [for all $n$-tuples $(x_1, \ldots, x_n)$ if $p$ is $n$-ary].

Let me try to get this straight. We can upgrade to a Galois *correspondence* if we take only the fixed points of the adjunction. These are subgroups of $G$ on one side, and subtheories of $P(X^*)$ on the other. The closure operators send collections of predicates to the their ‘theory’ closure, and collections of elements of $G$ to the subset of $G$ they generate.

Up the ladder we might hope for a (2)-connection which improves to a (2)-correspondence between 2-groups and 2-theories. So let’s recall what happens with a categorified connection:

each 2-relation $R: A^{op} \times B^{op} \to Set$ induces a pair of contravariant functors

$\frac{A^{'}(?) \to hom_{PB}(B^{'}(?),R(?, -))}{B^{'}(-) \to hom_{PA}(A^{'}(-),R(?,-))}$

[Just to help myself, I tried to think this out with $A$ and $B$ discrete. You get something like an isomorphism between

$\product_i (\product_j r_{ij}^{b_j})^{a_i}$ and $\product_j (\product_i r_{ij}^{a_i})^{b_j}.$

Apologies to any Australians reading for this horrible piece of concreteness.]

So where can we find a willing 2-relation to categorify the relation $R$, the set of $(g, p)$ such that $p(x_1, \ldots, x_n) = p(g x_1, \ldots, g x_n)$, all $n$-tuples $(x_1, \ldots, x_n)$?

In a single-sorted 2-theory we’ll have a category $C$ (or maybe a groupoid) instead of $X$, and then $A$ will presumably become the 2-group of automorphisms of $C$.

$B$ can become a category of functors from $C^n$ to Set for different values of $n$. So the sum over $n$ of the category of presheaves on $C^n$.

Then $R$ will need to be a functor from $A \times B$ (or perhaps their *op*s) to Set. So how will $R$ act on $(g, p)$, $g \in Aut(C)$ and $p: C^n \to Set$, to yield a set?

There’s plenty to remind one here of things glimpsed earlier – Kleinian 2-geometry, etc. – but my poor brain needs a rest right now.

## Re: 2-Galois and 2-Logic

Instead of saying

$(g, p) \in R iff p(x_1, \ldots, x_n) = p(g x_1, \ldots, g x_n), all (x_1, \ldots, x_n),$

we could define an action of $G$ on $P(X^n)$, so that $g \cdot p(x_1, \ldots, x_n) = p(g x_1, \ldots, g x_n)$.

Then

$(g, p) \in R iff g \cdot p = p.$

Up one level, we have $G = AUT(C)$ acting on $P(C^n)$. Now why not define

$R(g, p) = Isom_{P(C^n)}(p, g \cdot p),$

a set of isomorphisms of presheaves?