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.

The one place that equation (5) directly interacts with the output of Proposition 3.1 in a way that doesn’t just even out is Lemma 5.7, where (5) is checked for the weak map \psi\colon \mathfrak{g}_k\to \mathcal{P}_k\mathfrak{g} of 2-term L_\infty-algebras. There are flow-on effects, but this is the first place to look. In the proof of Lemma 5.7 the equation (5) is reduced to

Here \mathcal{P}_k\mathfrak{g} is strict, hence a 2-term dg-Lie algebra, arising from the crossed module of Lie algebras of the String crossed module. Note that there is a little bit of extra cancellation here that we will not assume below, so we will get more terms. And, it just so happens, Lada and Markl do give an explicit formulas for what weak maps from an L_\infty-algebra (in fact a general L_m-algebra) to a dg-Lie algebra:

From Lada–Markl, “Strongly homotopy Lie algebras”

Thus one can take Lada–Markl’s Definition 5.2, impose the fact m=\infty, that L and A are have underlying 2-term chain complexes, consider the case n=3 and take x_1,x_2,x_3\in L_0. The resulting equation directly gives rise to equation (5).

One thing to immediately note is that the elements \xi_{\tau(p)}=x_{\tau(p)} are all of degree zero, so that the factor (-1)^{(t-1)\sum_p|\xi_{\tau(p)}|} is always going to be +1. Also, the function f_3 has codomain a zero vector space, so the first term will vanish. We can thus reduce the equation above to

The funky sign factors \chi(\sigma),\chi(\tau) reduce here to just the signs of the permutations, since because for degree-zero terms the Koszul sign factor of the function \chi reduces to a +1. For the first outer sum, we can reduce to the cases k=2=j and k=3,\ j=1, because the function f_3 arising when j=3 vanishes. The sign (-1)^{k(j-1)}=+1 in all three cases. For the second outer sum we have s=1,\ t=2 and s=2,\ t=1.

So now we must do some checking to see which elements of S_3 are unshuffles of the appropriate sort, namely (1,2)-unshuffles, (2,1)-unshuffles and (3,0)-unshuffles, and which ones satisfy the additional constraint in the second row. HDA VI helpfully gives a list of the (1,2)-unshuffles, namely

Of these, only id satisfies \tau(1)<\tau(2). The (2,1)-unshuffles are id, (123) and (23), and the ones that satisfy \tau(1)<\tau(3) are the identity and (23). Lastly, the only (3,0)-unshuffle is the identity. From this we see that the left side of the equation has four terms, and the right side of the equation has three terms, which gives the expected seven terms (one of the terms in eq (5) from BCSS vanishes due to the strictness of A).

Writing out these sums we get the final equation:

or, accounting for the fact the bracket (from A) the bilinear function l_2 (from L), and f_2 are all antisymmetric on degree zero elements (as the x_i here are), we can rearrange this to be

This, apart from the missing l_3(f_1(x_1),f_1(x_2),f_1(x_3)) term (as it it is zero in this case), the coherence equation for a weak map of 2-term L_\infty-algebras, as it appears in Definition 4.2.4 of HDA VI, and not as equation (5) from BCSS.

To however, as far as verifying that \psi is a map of 2-term L_\infty-algebras, since the sign in the bracket action in the crossed module of Lie algebras arising from the String crossed module needs to be flipped, as per the previous two posts in this series, the proof of Lemma 5.7 holds up, since this sign flip means the corrected version of eq (5) will indeed hold! Flow-on effects arise, of course, since we need to ensure that \phi (Lemma 5.4) and \lambda (Lemma 5.5) are fixed up so as to also be maps of 2-term L_\infty-algebras. However, these can be fixed by simply flipping the sign on the functions $maps \phi_2$ and \lambda_2, as far as I can see. And this, it seems, is it. The contradiction from the definition of \beta_p is removed, the definition of (weak) L_\infty map is consistent with an example worked out in the cited source for all things L_\infty, and the main theorem still holds.

Aside from demanding the outright derivation of Lada–Markl’s Definition 5.2 from the generic, non-detailed definition of weak map in their Remark 5.3, I believe there is really nothing else one could do, here. My confidence is getting all the gradings and signs right is less than trusting the authors to be able to work through their own example, in this case. However, if anyone has any further ideas that expose why my argument in this series of blog posts is either flawed, or incomplete, I’d like to know. In particular, if someone has done the derivation of Definition 5.2 from Remark 5.3, or knows where it is done in the literature, then please give an answer at this MathOverflow question.

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