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July 4, 2007

The Inner Automorphism 3-Group of a Strict 2-Group

Posted by Urs Schreiber

David Roberts and I would like to share the following text:

D. Roberts, U.S.
The inner automorphism 3-group of a strict 2-group
arXiv:0708.1741


Abstract

Any group GG gives rise to its 2-group of inner automorphisms, INN(G)\mathrm{INN}(G). The nerve of this is the universal GG-bundle. Similarly, for every 2-group G (2)G_{(2)} there is a 3-group INN(G (2))\mathrm{INN}(G_{(2)}) of inner automorphisms. We construct this for G (2)G_{(2)} any strict 2-group, discuss how it can be understood as arising from the mapping cone of the identity on G (2)G_{(2)} and show that it fits into a short exact sequence Disc(G (2))INN(G (2))ΣG (2) \mathrm{Disc}(G_{(2)}) \to \mathrm{INN}(G_{(2)}) \to \Sigma G_{(2)} of strict 2-groupoids. We close by indicating how this makes INN(G (2))\mathrm{INN}(G_{(2)}) the universal G (2)G_{(2)}-2-bundle.

I am very glad that David Roberts joined me on the quest for inner peace – I mean: for inner automorphisms. I very much enjoyed this fruitful exchange. All by email, of course, and intercontinental, as befits a true blog community.

David has a bunch more notes on how everything and more works in the world of simplicial groups, making in particular the relations to Brano Jurčo’s work Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry more manifest.

We are grateful to Jim Stasheff for accompanying this capital INN(G (n))\mathrm{INN}(G_{(n)}) endeavour with his guidance, and for making us relate it, by the mapping cone construction, more closely to the lower case inn(g (n))\mathrm{inn}(g_{(n)}) business, which we have been involved in recently.


Related entries:

I started thinking about INN(G (2))\mathrm{INN}(G_{(2)}) in my first entry titled nn-Curvature. There the point was that we can understand the differential 2-cocycles given by Breen and Messing as morphisms Itracurv I \stackrel{\mathrm{tra}}{\to} \mathrm{curv} of 3-functors, where curv:P 3(X)ΣINN(G (2)) \mathrm{curv} : P_3(X) \to \Sigma \mathrm{INN}(G_{(2)}) plays the role of the 3-curvature of the pseudo transport 2-functor tra\mathrm{tra}, following the general mechanism described in Sections, States, Twists and Holography.

The trivializability of INN(G (2))\mathrm{INN}(G_{(2)}) is crucial here: it makes curv\mathrm{curv} a flat 3-transport, which is the Bianchi identity in this context, as described in the second entry called nn-Curvature as well as in Curvature, the Atiyah Sequence and Inner Automorphisms.

But this relation to 3-curvature is not mentioned in the present text. In fact, at the moment I am not completely sure how it relates conceptually to the fact – which we do discuss – that INN(G (2))\mathrm{INN}(G_{(2)}) plays the role of the universel G (2)G_{(2)}-2-bundle. It feels a litte embarrassing, since there ought to be an obvious relation. But at the moment it escapes me.

Σ(INN(G (2)))\Sigma(\mathrm{INN}(G_{(2)}))-3-transport is supposed to give the right notion of parallel transport in particular for spinning strings, being the integrated version of the 3-connections discussed in terms of Lie nn-algebras in Connections on String-2-Bundles, following the general mechanism described in Chern Lie (2n+1)-algebras.

(But notice that there seem to be also much more mundane applications: in The nn-Café quantum conjecture I indicated how ordinary U(1)U(1)-bundles with connection can be conceived in terms of INN()\mathrm{INN}(\mathbb{Z} \to \mathbb{R})-transport, thereby making the appearance of “phases” (i.e. elements in U(1)U(1)) a phenomenon emerging from more ordinary ingredienty like the integers and the reals. )

In fact, the realization described in Derivation Lie 1-Algebras of Lie nn-Algebras that the Lie (n+1)(n+1)-algebra inn(g (n)) \mathrm{inn}(g_{(n)}) of inner derivations of any Lie nn-algebra g (n)g_{(n)} arises as the mapping cone of the identity on g (n)g_{(n)} motivated David Roberts and myself, greatly influenced by Jim Stasheff’s guidance, to pursue what is now advertized as one of the two main results of our text: that INN(G (2))\mathrm{INN}(G_{(2)}) is indeed the nonabelian mapping cone, in the sense of Conduché, of the identity on G (2)G_{(2)}.

Finally, on a more speculative note, it seems remarkable that the construction of short exact sequences of Lie (2n+1)(2n+1)-algebras for every transgressive Lie (2n+1)(2n+1)-cocycle is very much controlled by the exact sequence Disc(G)INN(G)ΣG \mathrm{Disc}(G) \to \mathrm{INN}(G) \to \Sigma G of groupoids, for GG any Lie (1-)group, which plays the role of the universal GG-bundle.

This makes one naturally wonder if this entire construction has a counterpart when we replace the sequence Disc(G)INN(G)ΣG\mathrm{Disc}(G) \to \mathrm{INN}(G) \to \Sigma G by the sequence Disc(G (2))INN(G (2))ΣG (2) \mathrm{Disc}(G_{(2)}) \to \mathrm{INN}(G_{(2)}) \to \Sigma G_{(2)} for G (2)G_{(2)} any strict Lie 2-group, which plays the role of the universal G (2)G_{(2)}-2-bundle.

This should involve something like a categorification of the concept of a Lie algebra cocycle, namely a Lie 2-algebra cocycle, as well as the related notions of transgression and invariant polynomials.

I have the vague suspicion that this is what will finally allow to put the supergravity Lie 3-algebra in its proper home, using the fact that the Poincaré Lie algebra, which the supergravity Lie 3-algebra is an extension of, being a semidirect sum, really wants to be regarded as a strict Lie 2-algebra. But this I still haven’t figured out yet.

Posted at July 4, 2007 11:56 AM UTC

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Re: The inner automorphism 3-group of a strict 2-group

I should point out here one question, which we haven’t been able to fully clarify, but a discussion of which would be good:

as described at length, the 3-group INN(G (2))\mathrm{INN}(G_{(2)}) is essentially just the sub-3-group of the full autormorphism 3-group AUT(G (2))\mathrm{AUT}(G_{(2)}) containing all those automorphism which arise by conjugation with objects in G (2)G_{(2)}.

But not quite. The point is that we do distinguish in INN(G (2))\mathrm{INN}(G_{(2)}) two objects to be different, if they are different as objects of G (2)G_{(2)} but even if conjugating by them happens to lead to the same automorphism (so, if they differ by something “in the center”).

We describe explicitly how INN(G (2))\mathrm{INN}(G_{(2)}) is constructed. What is missing is a nice general 1-line abstract definition, which captures this funny little add-on, which says that we have to distinguish objects even if they have the same conjugation action.

In fact, in one dimension lower there is an indication for how it should work. But I can’t quite see how to generalize this:

There the point is this: for GG any ordinary group, we shouldn’t just look at automorphisms ΣGΣG \Sigma G \stackrel{\sim}{\to} \Sigma G and then demand them to be inner, etc., but we should just say this: look at the group of transformations of the form Id ΣG ΣG something \array{ & \nearrow \searrow^{\mathrm{Id}} \\ \Sigma G &\Downarrow& \Sigma G \\ & \searrow \nearrow^\sim_{\mathrm{something}} } under horizontal composition.

That solves the problem: the mere existence of such a transformation exactly constrains the lower automorphism to be inner. Moreover, the particular choice of transformation from the identity to that automorphism distinguishes between conjugations by different morphisms which lead to the same inner automorphism.

So, I was hoping to be able to say, strictly, that INN(G (n))\mathrm{INN}(G_{(n)}), for any nn-group G (n)G_{(n)} is something like the (n+1)(n+1)-group of transformations of the above form.

But

a) I am not entirely sure yet how to formulate this in general

b) while it gives the expected result for n=1n=1, I don’t see how it gives (up to equivalence), our desired construction (described at detail in the paper) of INN(G (2))\mathrm{INN}(G_{(2)}).

I’ll need to further think about this. If anyone has any comments on this, I’d be very grateful. Probably we are just missing the obvious.

Posted by: urs on July 6, 2007 7:17 AM | Permalink | Reply to this

Re: The inner automorphism 3-group of a strict 2-group

I’ll need to further think about this.

Now that thinking has been done. David and myself have an updated version of our document here.

This now contains the description of INN(G (2))\mathrm{INN}(G_{(2)}) in terms of tangent categories in section 4.

This allowed to give a nicer abstract description of INN(G (2))\mathrm{INN}(G_{(2)}) in proposition 3, allowing what is now section 5 to concentrate on spelling out what INN(G (2))\mathrm{INN}(G_{(2)}) looks like in terms of crossed modules of groups.

Ah, yeah, and to tease everybody I’ll say this: these inner automorphism nn-groups are related to holography by using this reformulation of the latter.

But that’s another story.

Posted by: Urs Schreiber on July 24, 2007 6:22 PM | Permalink | Reply to this
Read the post Tangent Categories
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Excerpt: An arrow-theoretic formulation of tangency and its relation to inner automorphisms.
Tracked: July 21, 2007 4:34 PM
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Excerpt: We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
Tracked: July 27, 2007 5:28 PM

Re: The Inner Automorphism 3-Group of a Strict 2-Group

As a reaction to a complaint, there is now a new version with an extended introduction, giving more motivation and more background.

If you feel it is still too boring (I am sure it is :-/) please drop us a note and complain.

Posted by: Urs Schreiber on August 1, 2007 9:45 PM | Permalink | Reply to this
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Tracked: August 10, 2007 6:21 PM

Re: The Inner Automorphism 3-Group of a Strict 2-Group

Now this thing has a preprint number:

http://arxiv.org/abs/0708.1741.

Posted by: Urs Schreiber on August 13, 2007 6:38 PM | Permalink | Reply to this

Re: The Inner Automorphism 3-Group of a Strict 2-Group

Ronnie Brown discusses something which is apparently a kind of generalization of what we considered: arXiv:0708.1677.

Posted by: Urs Schreiber on August 14, 2007 11:12 AM | Permalink | Reply to this
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Re: The Inner Automorphism 3-Group of a Strict 2-Group

David and I constructed the inner automorphism 3-group of the 2-group GG as a tangent category of GG, by mapping the “fat point”

pt={} \mathbf{pt} = \{\bullet \to \circ\}

into it. This was motivated by how tangents in synthetic differential geometry and, notably, in supergeometry come from mapping the “superpoint” into a space.

A more detailed discussion about how this is indeed quite analogous is now on p. 18-19 (the superpoint) and on p. 30-31 (the inner derivation Lie (n+1)(n+1)-algebra from maps from the superpoint) of L L_\infty-connections (pdf).

See in particular figure 3 on p. 30.

Posted by: Urs Schreiber on January 23, 2008 3:22 PM | Permalink | Reply to this
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