Questions on 2-covers

The following may well have been talked about by John in his lectures but I didn’t see it explicitly there, so I’ll ask.

Since at level 1, we have

A Galois connection between subgroups of the fundamental group ${\pi }_1(X)$ and path-connected covering spaces of X for path-connected $X$. A universal covering space is simply connected.

should we not expect a level 0 analogue:

A ‘connection’ between subsets of the set of connected components ${\pi }_0(X)$ and covering spaces of $X$ whose locally constant fibres are either empty or $\{*\}$?

This puts these latter ‘covering spaces’ into correspondence with the poset of subsets of ${\pi }_0(X)$, so that the universal covering space with truth valued fibres is the empty set.

So do we have then:

A Galois 2-connection between sub-2-groups of the fundamental 2-group ${\pi }_2(X)$ and path-connected 2-covering spaces (with groupoid fibres) of X for path-connected $X$, a universal 2-covering space being 2-connected?

So, if we think of a nice space with nontrivial first and second homotopy, say the loop space of the 2-sphere, do we have a correspondence between 2-covering spaces and sub-2-groups of its fundamental 2-group? And is there a universal 2-connected 2-cover with 1- and 2-homotopy killed off?

Returning to Lautman

I mentioned in an earlier post that Albert Lautman had a considerable influence on my decision to turn to philosophy. I recently found out that his writings have been gathered together and republished as Les mathématiques, les idées et le réel physique, Vrin, 2006, a copy of which arrived through the post the other day. It’s remarkable how much contemporary mathematics Lautman covers – class field theory, algebraic topology, analytic number theory, etc.

At the same time as I was reading Lautman I became excited by category theory, via Colin McLarty’s Uses and Abuses of the History of Topos Theory, British Journal for the Philosophy of Science 1990 41(3):351-375, and Saunders Mac Lane’s Mathematics: Function and Form, and noticed an affinity with Lautman’s thinking, supported by a remark made by Jean Dieudonné in his 1977 Preface:

La “montée vers l’absolu” qu’il y discerne, et où il voit une tendance générale, a pris en effet, grâce au langage des catégories, une forme applicable à toutes les parties des mathématiques: c’est la notion de ‘foncteur représentable’ qui joue aujourd’hui un rôle considérable, tant dans la découverte que dans la structuration d’une théorie. (p. 36)

That Lautman worked with Claude Chevalley and Charles Ehresmann may not be unconnected.

The 2006 edition includes an interesting introduction – Lautman et la dialectique créatrice des mathématiques – by Fernando Zalamea, who devotes a section to interpreting Lautman through category theoretic spectacles. In successive posts I’ll take a look at how amenable are his mathematical examples to this treatment.

I’ll end now with a quotation Zalamea has selected from Lautman, which nicely expresses what might be done instead of Anglophone philosophy of mathematics:

La philosophie mathématique, telle que nous la concevons, ne consiste donc pas tant à retrouver un problème logigue de la métaphysique classique au sein d’une théorie mathématique, qu’à appréhender globalement la structure de cette théorie pour dégager le problème logique qui se trouve à la fois défini et résolu par l’existence même de cette théorie.

Yes, to what problems or questions is mathematics a response?

Dual Formulation of String Theory and Fivebrane Structures

We would like to share the following:

Hisham Sati, U.S. and Jim Stasheff

Dual Formulation of String Theory and Fivebrane Structures

(pdf)

Abstract.

We study the cohomological physics of fivebranes
in type II and heterotic string theory. We give an interpretation of
the one-loop term in type IIA, which involves the first and second
Pontrjagin classes of spacetime, in terms of obstructions to
having bundles with certain structure groups.
Using a generalization of the Green-Schwarz anomaly
cancelation in heterotic string theory which demands the target space
to have a String structure,
we observe that the “magnetic dual” version of
the anomaly cancellation condition
can be read as a higher analog of String structure,
which we call Fivebrane structure.
This involves lifts of orthogonal and unitary structures
through higher connected covers which are not just
3- but even 7-connected.
We discuss the topological obstructions to the existence
of Fivebrane structures.
The dual version of the anomaly cancelation
points to a relation
of String and Fivebrane structures under electric-magnetic duality.

This expands on some of the material announced in section 3 of

H. S., U.S., J. S.

$L_{\mathrm{\infty }}$-connections and application to String- and Chern-Simons transport

(arXiv, blog pdf)

but so far concentrates on the topological aspects of Fivebrane structures. A discussion of the differential geometry of Fivebrane 6-bundles with connection – which are nonabelian differential cocycles that are to super 5-branes as String 2-bundles with connection are to superstrings and as ordinary Spin bundles with connection are to spinning particles – as well as of the Chern-Simons 7-bundles with connection obstructing their existence, will be given elsewhere, following the general approach described in

On nonabelian differential cohomology

(pdf).

I’d be grateful for comments, but should add that I’ll be travelling in Ireland until 3rd of May, which will reduce my responsiveness here for that period.