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Started at 2-groupoid of Lie 2-algebra valued forms a section for general Lie 2-algebras .
(Intentionally all in unreadable components. I am preparing material for a communication with some physicists ;-)
How about groupoid of Lie algebra valued forms? I know, I’m thinking negatively here. :-)
This sequence of entries here exists, if that’s what you are alluding to:
(There was something wrong with the redirects to the first one, though. I have fixed it now.)
Ah, inconsistent hyphens! Thanks, I see it now.
Yes, sorry, it’s my fault. I have made dozens of different version of writing these entry titles be redirects. Probably all of them wrong! And then I missed the single right one, I guess. ;-)
I made a correction in the condition that 2-arrows need to satisfy, in the section about strict Lie 2-algebras: there was an $a$ that needed to be an $A$. I also made some reference to compositions of 2-arrows there that had been left as an ellipsis for some time…
What’s a good reference in the literature for the definition of the 2-groupoid of Lie 2-algebra valued forms for a strict Lie 2-algebra as presented here without the fake flatness conditions (the main definition on this page)? In the listed references on the page and elsewhere there are similar 2-groupoids for flat forms presented, but I can’t find any reference to the more general stack. Might one expect this more general stack to be equivalent to the stack that modulates Lie 2-algebra valued forms using differential forms on products of smooth simplices and intersections (i.e. $\mathrm{exp}_{\mathrm{conn}}(\mathfrak{g})$ as presented in Cech cocycles for differential characteristic classes)?
If there is no flatness condition then one needs to decide what (higher) gauge transformations one wants to incorporate in the 2-gorupoid.
For $\mathfrak{g}$ Lie 2-algebra, the unconstrained $\mathfrak{g}$-valued forms as just
$Hom_{dgAlg}\big(W(\mathfrak{g}),\,\Omega^\bullet_{dR}(-)\big) \;\simeq\; Hom_{Ch}\big(\mathfrak{g},\,\Omega^\bullet_{dR}(-)\big) \;\simeq\; \big(\Omega^\bullet_{dR}(-) \otimes \mathfrak{g}\big)_0$where $W(-)$ denotes the Weil algebra
(and assuming finite-dimensionality of $\mathfrak{g}$ in the last step).
This is to be contrasted with the case of flat forms, which is
$Hom_{dgAlg}\big(CE(\mathfrak{g}),\,\Omega^\bullet_{dR}(-)\big) \;\equiv\; \Omega_{dR}(-, \mathfrak{g})_{flat} \,.$where now $CE(-)$ denotes the Chevalley-Eilenberg algebra.
So one natural choice for the 2-groupoid of unconstrained forms is the evident analog of the flat case, which is given by the simplicial presheaf
$(U \in CartSp, \, [k] \in \Delta) \;\mapsto\; Hom_{dgAlg}\big(CE(\mathfrak{g}),\,\Omega^\bullet_{dR}(U \times \Delta^k)\big)$namely now given by the analogous simplicial presheaf with “$CE$” replaced by “$W$”
$(U \in CartSp, \, [k] \in \Delta) \;\mapsto\; Hom_{dgAlg}\big(W(\mathfrak{g}),\,\Omega^\bullet_{dR}(U \times \Delta^k)\big) \,.$This is implicitly what appears in “Cech cocycles for differential characteristic classes” (there subjected two the two “Ehresmann conditions”).
It may be useful to think of this the following way:
One can regard the Weil algebra of any Lie $n$-algebra $\mathfrak{g}$ as being the CE-algebra of the “inner derivation Lie $(n+1)$-algebra” $inn(\mathfrak{g})$.
$W(\mathfrak{g}) \,=\, CE\big(inn(\mathfrak{g})\big)$(eg. (131) on p 31 here).
This allows to regard non-flat $n$-connections with coeffcients in $\mathfrak{g}$ as flat $n+1$-connections with coefficients in $inn(\mathfrak{g})$ (expanded on for the simplest case $n = 1$ on pp 22 here): The curvature of the $n$-connection becomes a component of the $(n+1)$-connection and the flatness of the latter is just the “Bianchi identities” on the (unconstrained) curvatre components of the former.
$\,$
For a more component-based reference, maybe have a look at Martins & Picken (2011) (though I admit I would have to remind myself of what exactly is happening there).
9 Typo Chevallet, so the link does not work.
Thanks. Fixed now.
Hi Urs! Thank you for your reply! I think my previous question was a bit unclear.
In your answer you refer to the 2-groupoids of forms that are given by the Lie integration stacks; these simplicial presheaves of maps into the dg-algebra of differential forms on $U\times\Delta^{k}$.
In your paper with Konrad, “Smooth functors vs differential forms”: https://arxiv.org/pdf/0802.0663.pdf#page=29, is the definition of the 2-category $Z_{X}^{2}(\mathfrak{G})^{\infty}$. This looks like the 2-groupoid of flat Lie 2-algebra valued forms which is also referenced on this page (in the first proposition under section 3). In the definition of $Z_{X}^{2}(\mathfrak{G})^{\infty}$, forms on smooth simplices do not appear. On this page it’s stated that $Z_{X}^{2}(\mathfrak{G})^{\infty}$ is a subcategory of what’s here called $\overline\mathbf{B}G$, which looks to me like a 2-groupoid of Lie 2-algebra valued forms that doesn’t impose the fake flatness conditions. $\overline{\mathbf{B}}G$ is also constructed (on this page) without mention of forms on smooth simplices.
I was wondering if there was anywhere in the literature that $\overline{\mathbf{B}}G$ is mentioned (in this particular incarnation without any mention of forms on $\Delta^{k}$), or if it was someone’s original contribution to the nLab and it’s not yet present in the literature.
The reason I’m asking is because I’m wondering whether ultimately $\overline{\mathbf{B}}G$ and $Z_{X}^{2}(\mathfrak{G})^{\infty}$ are equivalent (in some appropriate sense) to the Lie integration stacks which do employ forms on $\Delta^{k}$ in their definition.
It just dawned on me now that perhaps my confusion is coming from a typo! Is $\overline{\mathbf{B}}G$ intended to be the same object as $Z^{2}_{X}(\mathfrak{G})$? I initially confused $Z^{2}_{X}(\mathfrak{G})$ for the stack of flat forms, but only one of the two flatness conditions are imposed on the data in $Z^{2}_{X}(\mathfrak{G})$: $dA + [A\wedge A] = t_{*}B$, and the second is not explicitly imposed.
If $Z^{2}_{X}(\mathfrak{G})$ is the stack of general (not necessarily flat) Lie 2-algebra forms, and $\overline{\mathbf{B}}G$ is intended to denote the same object, then this above condition is missing from the page in the definition of $\overline{\mathbf{B}}G$ and should be added (I could do this myself after some confirmation).
Let’s see.
The condition $F_A = t_\ast(B)$ (p. 29) is the first component of the flatness condition (aka “fake flatness”).
This condition is necessary (only) in order to make the 2-groupoid of Lie-algebra value 2-forms be identified with that of smooth 2-functors $P_2(-) \to \mathbf{B}G$,
which is why it is considered in arXiv:0802.0663.
However, it should not be necessary just to make the 2-groupoid of Lie 2-algebra value forms.
So I think it’s not a typo in the entry.
On the other hand, there is loads of room to improve exposition and notation of this ancient and neglected entry.
To start with, the overline notation is clearly not suggestive of anything, it would be good to change to something more suggestive.
Next, the issue that you are grappling with should actually be explained in the entry:
There is
(1.) the unconstrained 2-groupoid of forms, as stated,
(2.) its sub-groupoid (speaking strictly) with the “fake” curvature condition imposed and
(3.) the further sub-groupoid with both curvature constraints imposed.
If you have the energy to work on beautifying this entry, please do! That would be a great service to the community.
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