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February 13, 2008

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 RA×BR \subseteq A \times B whatsoever can be used to set up a Galois connection between subsets AA' of AA and subsets BB' of BB: A{aA: b:B(a,b)R}B{bB: a:A(a,b)R}iff\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, AA is the group of permutations gg on XX, BB is the set of finitary relations pp on XX, and the relation RR is the set of (g,p)(g, p) such that p(x 1,,x n)=p(gx 1,,gx n)p(x_1, \ldots, x_n) = p(g x_1, \ldots, g x_n) [for all nn-tuples (x 1,,x n)(x_1, \ldots, x_n) if pp is nn-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 GG on one side, and subtheories of P(X *)P(X^*) on the other. The closure operators send collections of predicates to the their ‘theory’ closure, and collections of elements of GG to the subset of GG 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×B opSetR: A^{op} \times B^{op} \to Set induces a pair of contravariant functors

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

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

i( jr ij b j) a i \product_i (\product_j r_{ij}^{b_j})^{a_i} and j( ir ij a i) b j. \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 RR, the set of (g,p)(g, p) such that p(x 1,,x n)=p(gx 1,,gx n)p(x_1, \ldots, x_n) = p(g x_1, \ldots, g x_n), all nn-tuples (x 1,,x n)(x_1, \ldots, x_n)?

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

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

Then RR will need to be a functor from A×BA \times B (or perhaps their ops) to Set. So how will RR act on (g,p)(g, p), gAut(C)g \in Aut(C) and p:C nSetp: 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

Instead of saying

(g,p)Riffp(x 1,,x n)=p(gx 1,,gx n),all(x 1,,x n), (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 GG on P(X n)P(X^n), so that gp(x 1,,x n)=p(gx 1,,gx n)g \cdot p(x_1, \ldots, x_n) = p(g x_1, \ldots, g x_n).

Then

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

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

R(g,p)=Isom P(C n)(p,gp), 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

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 PCP C to refer to Set C opSet^{C^{op}}, for a category CC, just as we use PXP X to refer to 2 X2^X, for a set XX.

If a subgroup of GG is a kind of member of PUGP U G, the power set of the underlying set of GG, 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

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

But you can still classify subobjects with the old classifier 1Ω1\to\Omega (where Ω\Omega is the category of subsets of 11, i.e. the order of truth values*). There is an equivalence between functors UGΩUG\to \Omega (i.e presheafs which take value in ΩSet\Omega\hookrightarrow\mathrm{Set}) and full and faithful functors CUGC\to UG, where CC is a groupoid. So the subobjects corresponding to predicates with truth value in Ω\Omega are the full and faithful functors.

* I don’t call it 22 because I prefer not to suppose that Set\mathrm{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

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 XX and we’re going to put into correspondence single-sorted XX-theories, and subgroups of X!X !, permutations of XX.

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!X !. But of course this isn’t so. If X={a,b,c}X = \{a, b, c\}, then unary predicates aren’t enough for a theory invariant under A 3A_3, even permutations. Binary predicates, however, are enough.

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

So, here we pack together all powers of XX, and then look at collections of maps to 22, 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 XX? 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)\Pi_1 (S^1), which has a 2-group acting on it, AUT(Π 1(S 1))AUT (\Pi_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

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 PPCP 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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