Correction to the definition of the String crossed module, part 4: the coherence equation

In the last post I showed that the paper BCSS contains a contradictory definition, namely of a particular 1-form \beta_p, and that flipping the sign out the front fixes it up. now I need to make sure that I ensure the main theorem is still valid, since it implicitly depends on that sign, through the conclusion of Proposition 3.1. My claim in the first post in this series is that the issue is hiding in equation (5) of BCSS, whereas Urs Schreiber cautioned me that the equation should be checked against the L_\infty-algebra literature, rather than just aiming for internal consistency. In particular the paper “Strongly homotopy Lie algebras” by Lada–Markl (arXiv), which is the cited source for the definition and conventions for L_\infty-algebras. The definition that needs to be used is that of ‘weak map’, except the description is slightly indirect, and there are several layers of interacting sign conventions that I felt I was going to come unstuck on. In this post I will deal with my workaround to arrive at what I hope is a definitive answer.

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Correction to the definition of the String crossed module, part 3: arriving at the contradiction, and the fix

Continuing on from the previous post, I need to finish off the calculation that shows the definition of the 1-form

\beta_p = -2\int_0^{2\pi} \langle \Theta,p(\theta)^{-1}\partial_\theta p(\theta)\rangle\,d\theta \in \Omega^1(P\Omega G)

in BCSS leads to a contradiction, for G a suitable compact connected Lie group, taken to be SU(2) for concreteness. Recall that I had calculated the difference between a pair of connections on the U(1)-bundle \widehat{\Omega SU(2)}\to \Omega SU(2), one arising from the construction of Murray and Stevenson, and the other being the pullback of this along the lift of Ad_p\colon \Omega SU(2) \to \Omega SU(2) for p a path in SU(2) starting at the identity matrix. This lift is defined in BCSS using \beta_p. In actual fact, we did the calculation on the trivial bundle over P\Omega SU(2) that is the pullback of the one I care about, and arrived at the formula

ev_{2\pi}^*(\widehat{Ad}_p^*\mu - \mu - \beta_p) = 2\int_{[0,2\pi]}ev^*(d\beta_p)

where ev_{2\pi}\colon P\Omega SU(2)\to \Omega SU(2) evaluates a path at the endpoint. I can show that for a careful choice of p, path f\colon [0,2\pi]\to \Omega SU(2) and tangent vector at f that is vertical for ev_{2\pi}, the right hand side fails to vanish. The left hand side is identically zero for vertical tangent vectors, and this will be the contradiction.

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Correction to the definition of the String crossed module, part 2: triangulation

After the previous post was shared on Twitter, and I tagged Urs Schreiber, one of the authors of the paper in question (‘BCSS‘), he rightly pointed out that I had merely found a discrepancy in the coherence equations between BCSS and HDA VI. The true source of equation (5) of BCSS (which I claimed last time was in error) should be, he says, the theory of weak maps of L_\infty algebras, going back to Lada and Markl’s Strongly homotopy lie algebras (or free arXiv version). I totally agree with this point, but I felt daunted by having to unwind the various definitions that unravel to equation (5). So I wanted to explore the issue a bit more. I know that the sign in front of the 1-form

\beta_p = -2\int_0^{2\pi} \langle \Theta,p(\theta)^{-1}\partial_\theta p(\theta)\rangle\,d\theta

in the proof of Proposition 3.1 cannot be correct, since it leads to a contradiction in a geometric calculation independent of all the L_\infty-algebra material. I should note that this geometric calculation is nothing exotic; a correspondent let me know I muddied the water in the previous post by discussing it in the context of a larger project with more novel constructions. As a result, I will give the calculation later below the fold.

What I did not realise when I wrote the first post is that the sign error in \beta_p is linked not only the an overall sign in equation (5) of BCCS, but also to a plus sign implicit in the definition of the comparison functor from 2-term L_\infty-algebras to (semistrict) Lie 2-algebras, which are possibly weak Lie algebra objects in the 2-category of groupoids internal to \mathbf{Vect}. This functor is constructed in HDA VI, in Theorem 4.3.6 (=Theorem 36 in the arXiv version), and here is the relevant version

Part of the proof of Theorem 4.3.6 of HDA VI, constructing a functor from 2-term L_\infty-algebras to Lie 2-algebras

This was invisible to me, until I checked a different article generalising HDA VI, namely Roytenberg’s On weak Lie 2-algebras (or free arXiv version), where a generalisation of the comparison functor to weaker objects is given. Here is included a minus sign on what turns out to be the troublesome piece of data arising from the crossed module of Lie algebras, arising from the sign on \beta_p.

Discussion in Roytenberg about the relation between weak 2-term L_\infty-algebra maps and morphisms between weak Lie 2-algebras

So now in fact one can see the sign on \beta_p as being a combination of two signs: one from the comparison functor, and one from the coherence equation (5), just that in BCSS, the first of these is a +, and the second is a - .

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Diophantine fruit

I guess people may be familiar with the sorts of memes with equations involving fruit that are generally relatively trivial, while claiming something like “95% of people can’t solve this!” Sometimes this leads to amusing parodies. However someone went the whole other way and created an real, solvable instance of the problem using only natural numbers whose smallest solution is … very large.

I like the way this type of problem can actually open up a dialogue of how mathematics is really open ended, creative and unfinished. Framing a fruit equation problem as a filter that makes people either feel stupid for not understanding it (it is basically algebra, after all), or smug and superior for getting it, is extremely unhelpful. But explaining that what can seem like an easy problem actually requires a lot of work, and would stump professional mathematicians (cf Fermat’s last theorem), is a good conversation starter. It can also touch on things like how geometry is brought to bear on algebraic problems (and vice versa!), how the solution can use methods not present in the problem and so on.

So, in honour of this, I thought I take what seems to be, at time of writing, the simplest currently unsolved Diophantine equation (that is: a multivariable polynomial, and looking only for whole-number solutions), and turn it into a fruit equation. We can think of it as trying to count fruit:

Here ‘simplest’ is according to the notion of “size” defined in this MathOverflow question, basically it’s a measure of how large all the powers and coefficients of a multivariable polynomial is. There are only finitely many polynomials of a given size. The polynomial from the picture is -x^3 + y^2 + z^2 - xyz + 5, and has size 29. Every polynomial of size 28 and smaller has either been solved or shown to have no solutions. The idea is to see (experimentally) where the rough threshold is between equations than be solved in a more-or-less elementary way, and where really serious techniques or obstructions really kick in, of the sort outlined here.

I’m happy for people to share the above image as widely as they can. If nothing else, maybe someone will actually solve the equation above, and contribute a piece of new knowledge!

Correction to the definition of the String crossed module

(See also part 2)

I have been very slowly working, with my colleague Raymond Vozzo, on a project that involves a very explicit calculation for bundle gerbes, and in particular with the gerbe underlying the crossed module of Lie groups introduced by Baez, Crans, Schreiber and Stevenson (BCSS), which presents the String 2-group \mathrm{String}_G of a suitable compact simply-connected Lie group G. However, there was a particular step where we got thoroughly stuck, where a certain calculation involving the comparison of two explicitly described connections on (a pullback of) the universal central extension U(1) \to \widehat{\Omega G} \to \Omega G of Fréchet Lie groups.

Aside from needing to overcome a sign convention around the Stokes theorem for integration along a fibre of a family of manifolds with boundary, there was still an issue where I got a formula that implied a certain non-zero 2-form was equal to its own negative. The origin of this negative sign is, ultimately, the minus sign in the definition of a certain family of 1-forms \beta_p on \Omega G appearing in the proof of Proposition 3.1 in BCSS. In my own notation it is

\beta_p = -2\int_0^{2\pi} \langle \Theta,p(\theta)^{-1}\partial_\theta p(\theta)\rangle\,d\theta

where p\colon [0,2\pi]\to G is a smooth path starting at the identity element, \Theta is the left-invariant Maurer–Cartan form on \Omega G and \langle \cdot , \cdot\rangle is a suitably normalised version of the (pointwise-evaluated) Killing form on \mathfrak{g}. So I really couldn’t figure out what was going wrong, since I was fairly confident in our computations. This computation fed into a much larger computation, where if I ignored this problem and pressed on, trusting my instincts, things seemed to be going smoothly, and reproducing something like what we’d expect. I also had another (slow-burn) project involving this same comparison of connections, where the sign issue was also arising. The theory there pointed in one direction, the computation in another.

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The strange adventure of the Universal Coefficient Theorem in the night

If you trawl the internet for more exotic Universal Coefficient Theorems, then you’ll come across comments to the effect that there isn’t a UCT for cohomology with local coefficients in general (for instance, here, for group cohomology, which is ordinary cohomology of a K(G,1), or the Groupprops wiki page, which only gives the trivial coefficients version). I had reason to want a UCT for group cohomology with coefficients a nontrivial module, to try to clear some hypotheses using formal properties of group homology (it’s better behaviour with respect to filtered colimits, in particular). The only reference I could find for a suitable UCT was an exercise in Spanier’s venerable book:

Page 283 of Spanier’s book Algebraic Topology

I found a reference in a 2018 paper, that said “there is a version [of the local coefficient UCT] in [Spanier], p. 283, though its application is limited”. Up until this point, I had not actually seen the statement written out anywhere! In particular, it’s not clear what Spanier’s assumptions are (it might be he is assuming R is a PID throughout this section, but I couldn’t see it on a quick search), and in particular, something must break for this to be of “limited” application.

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This argument of Mochizuki doesn’t make sense to me

I made this point in a comment at Not Even Wrong, but I think it worth amplifying. It is regarding section 2.3 of a recent document released by Mochizuki, which I reproduce below the fold, along with my discussion. I have taken a screenshot as I don’t know if the document will be updated or not. I’m trying to figure this out as I go, it’s just so bizarre [Edit as in, I was literally thinking it through as I wrote this, and I didn’t edit it, on purpose. It’s what David Butler tags as #trymathslive on Twitter].

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Algebraic Topology: First Steps in Cohomology — videos available now!

tl;dr Here’s a free series of 24 lectures on algebraic topology!

In January and February this year, I taught an intensive course for the Australian Mathematical Sciences Institute (AMSI), as part of the annual summer school program. It usually rotates around the country, and students would normally all travel to the host university, and live and study together for four weeks. The course counts for a full semester’s credit towards their honours or masters coursework (I guess in the US system it would be roughly equivalent to a first semester grad course). However, this year things were … different, so the summer school was held online, though still hosted by the University of Adelaide, with lecturers (as usual) from around the country teaching (I ran my lectures on Twitch, inspired by Signum University).

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Algebraic topology class memes

Over the summer I taught a four-week intro graduate-level class on algebraic topology, with a focus on cohomology and homological algebra, using mostly combinatorial methods until the end, when we did singular cohomology, briefly. I encouraged my students to have a little fun in the class Discord server, and meme away. Here are some they come up with, plus one from me. For some reason, the little squiggly picture, which is meant to be the terminal 2-skeletal \Delta-set and that I called P, got the nickname ‘Pete’. The terminal \Delta-set, which I called P_\infty, got called “infinite Pete”. Some of these I don’t even know what they were in reaction to. But the skeleton one I know is because my frame rate was super super low before I got the OBS settings right, and the lag was … something else.