2-Galois and 2-Logic

February 13, 2008

Posted by David Corfield

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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 $p (x_{1}, \dots, x_{n}) = p (g x_{1}, \dots, g x_{n})$ [for all $n$ -tuples $(x_{1}, \dots, x_{n})$ if $p$ is $n$ -ary].

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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

$\prod_{i} (\prod_{j} r_{ij}^{b_{j}})^{a_{i}}$ and $\prod_{j} (\prod_{i} r_{ij}^{a_{i}})^{b_{j}} .$

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 $p (x_{1}, \dots, x_{n}) = p (g x_{1}, \dots, g x_{n})$ , all $n$ -tuples $(x_{1}, \dots, x_{n})$ ?

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 ops) 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.

Posted at February 13, 2008 9:44 PM UTC

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Re: 2-Galois and 2-Logic

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Instead of saying

$(g, p) \in R iff p (x_{1}, \dots, x_{n}) = p (g x_{1}, \dots, g x_{n}), all (x_{1}, \dots, x_{n}),$

we could define an action of $G$ on $P (X^{n})$ , so that $g \cdot p (x_{1}, \dots, x_{n}) = p (g x_{1}, \dots, g x_{n})$ .

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?

Posted by: David Corfield on February 14, 2008 11:21 AM | Permalink | Reply to this

Re: 2-Galois and 2-Logic

Wow. Now ‘categorified connection’ means two really different things on this blog!

(If anyone explains how Galois connections and connections on bundles are special cases of a more general concept, I’ll kill ‘em.)

Posted by: John Baez on February 14, 2008 9:25 PM | Permalink | Reply to this

Re: 2-Galois and 2-Logic

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A long time ago (in an old Klein 2-geometry thread) we were having trouble deciding how to define a sub-2-group. Could it be that this was because we didn’t think back then about how to categorify the power set construction?

Todd has told us that people use $P C$ to refer to ${Set}^{C^{op}}$ , for a category $C$ , just as we use $P X$ to refer to $2^{X}$ , for a set $X$ .

If a subgroup of $G$ is a kind of member of $P U G$ , the power set of the underlying set of $G$ , should a sub-2-group of a 2-group be a kind of presheaf on the underlying category of the 2-group?

Then again, we might have told a similar story already for groups themselves, which are after all categories. So that, as Lawvere said, a presheaf on a group is a predicate.

Posted by: David Corfield on February 18, 2008 9:31 AM | Permalink | Reply to this

Re: 2-Galois and 2-Logic

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Since a 2-group is a groupoid, there is a simplification in the classification of subobjects : there is an equivalence between presheafs $UG \to Set$ and faithful functors $C \to UG$ , where $C$ is a groupoid. So the subobjects given by predicates with “truth value” in $Set$ are the faithful functors into $G$ (in the 2-category of groupoids).

But you can still classify subobjects with the old classifier $1 \to Ω$ (where $Ω$ is the category of subsets of $1$ , i.e. the order of truth values*). There is an equivalence between functors $UG \to Ω$ (i.e presheafs which take value in $Ω ↪ Set$ ) and full and faithful functors $C \to UG$ , where $C$ is a groupoid. So the subobjects corresponding to predicates with truth value in $Ω$ are the full and faithful functors.

* I don’t call it $2$ because I prefer not to suppose that $Set$ is boolean, but it’s another question.

Posted by: Mathieu Dupont on February 18, 2008 12:19 PM | Permalink | Reply to this

Re: 2-Galois and 2-Logic

Maybe, as John once said, we’ll end up with split forms of categorified notions such as sub-2-groups, depending on use.

Posted by: David Corfield on February 19, 2008 6:37 PM | Permalink | Reply to this

Re: 2-Galois and 2-Logic

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How does the Galois theory of deck transformations and covering spaces fit in with what’s being discussed here.

In the Trimble-Dolan setting, we have a set $X$ and we’re going to put into correspondence single-sorted $X$ -theories, and subgroups of $X!$ , permutations of $X$ .

You might at first hope that unary predicates would be enough to pin down the set members sufficiently to restrict allowable symmetries to a given subset of $X!$ . But of course this isn’t so. If $X = {a, b, c}$ , then unary predicates aren’t enough for a theory invariant under $A_{3}$ , even permutations. Binary predicates, however, are enough.

[Is there a general result as to how small the largest $n$ need be such that all subgroups of $m!$ are captured by theories with predicates of arity no more than $n$ ?]

So, here we pack together all powers of $X$ , and then look at collections of maps to $2$ , or truth-valued fibrations above it.

When we turn to covering spaces, then, say of the circle, why aren’t we seeing powers of the circle, just as we saw powers of $X$ ? Perhaps it’s because this situation is already 2-Galois. After all, we probe this circle by looking at locally constant sheaves on it, which seems to be a rather like we’re treating it as a category.

Say we start with the fundamental groupoid of the circle, $Π_{1} (S^{1})$ , which has a 2-group acting on it, $AUT (Π_{1} (S^{1}))$ . My attempt to find a 2-correspondence was to look to (pre)-sheaves over powers of the former and representations of the latter.

Powers again.

Posted by: David Corfield on February 18, 2008 11:10 AM | Permalink | Reply to this

Re: 2-Galois and 2-Logic

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I seem to be rambling on by myself here, but a couple of thoughts. Were the space not connected I could see we might need locally constant sheaves on powers of the space.

As a reminder, the thought that we ought to have something to do with $P P C$ in 2-Galois relates to the Grothendieckian take on fundamental groups as the symmetries of a map from the category of locally constant sheaves on a space to Set.

Posted by: David Corfield on February 19, 2008 6:45 PM | Permalink | Reply to this

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February 13, 2008