∞-Lie theory (higher geometry)
For $\mathfrak{g}$ a Lie 2-algebra the 2-groupoid of $\mathfrak{g}$-valued forms is the 2-groupoid whose objects are differential forms with values in $\mathfrak{g}$, whose morphisms are gauge transformations between these, and whose 2-morphisms are higher order gauge transformations of those.
This naturally refines to a non-concrete Lie 2-groupoid is the 2-truncated ∞-Lie groupoid whose $U$-parameterized smooth families of objects are smooth differential forms with values in a Lie 2-algebra, and whose morphisms are gauge transformations of these.
This is the higher category generalization of the groupoid of Lie-algebra valued forms.
A cocycle with coefficients in this 2-groupoid is a connection on a 2-bundle.
Consider a Lie strict 2-group $G$ corresponding to a Lie crossed module $(G_2 \stackrel{\delta}{\to} G_1)$ with action $\alpha : G_1 \to Aut(G_2)$. Write $\mathbf{B}G$ for the corresponding delooping 2-groupoid, the one coming from the crossed complex
Write $[\mathfrak{g}_2 \stackrel{\delta_*}{\to} \mathfrak{g}_1]$ for the corresponding differential crossed module with action $\alpha_* : \mathfrak{g}_1 \to der(\mathfrak{g}_2)$
The 2-groupoid of Lie 2-algebra valued forms is defined to be the 2-stack
which assigns to $U \in CartSp$ the following 2-groupoid:
An object is a pair
A 1-morphism $(g,a) : (A,B) \to (A',B')$ is a pair
such that
and
The composite of two 1-morphisms
is given by the pair
a 2-morphism $f : (g,a) \Rightarrow (g', a'):(A,B)\to (A',B')$ is a function
such that
and
and composition is defined as follows: vertical composition is given by pointwise multiplication (DR: the order still needs sorting out!) and horizontal composition is given as horizontal composition in the one-object 2-groupoid $\mathbf{B}G)$.
We consider now $\mathfrak{g}$ a general Lie 2-algebra.
Let $\mathfrak{g}_0$ and $\mathfrak{g}_1$ be the two vector spaces involved and let
be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general Chevalley-Eilenberg algebra
with these generators.
We thus have
for collections of structure constants $\{C^a{}_{b c}\}$ (the bracket on $\mathfrak{g}_0$) and $\{r^i_a\}$ (the differential $\mathfrak{g}_1 \to \mathfrak{g}_0$) and $\{\alpha^i{}_{a j}\}$ (the action of $\mathfrak{g}_0$ on $\mathfrak{g}_1$) and $\{r_{a b c}\}$ (the “Jacobiator” for the bracket on $\mathfrak{g}_0$).
These constants are subject to constraints (the weak Jacobi identity and its higher coherence laws) which are precisely equivalent to the condition
Over a test space $U$ a $\mathfrak{g}$-valued form datum is a morphism
from the Weil algebra $W(\mathfrak{g})$.
This is given by a 1-form
and a 2-form
The curvature of this is $(\beta, H)$, where the 2-form component (“fake curvature”) is
and whose 3-form component is
(flat Lie 2-algebra valued forms)
The full sub-2-groupoid on flat Lie 2-algebra valued forms, i.e. those pairs $(A,B)$ for which the 2-form curvature
and the 3-form curvature
vanishes is a resolution of the underlying discrete Lie 2-groupoid $\mathbf{\flat} \mathbf{B}G$ of the Lie 2-groupoid $\mathbf{B}G$.
This is discussed at ∞-Lie groupoid in the section strict Lie 2-groups – differential coefficients.
Let $\mathbf{\Pi}_2 : CartSp \to 2LieGrpd$ be the smooth 2-fundamental groupoid functor and let $P_2 : CartSp \to 2LieGrpd$ be the path 2-groupoid functor, taking values in the 2-catgeory $2Grpd(Difeol)$ of 2-groupoids internalization to diffeological spaces. Then
the 2-groupoid of Lie 2-algebra valued forms for which both 2- and 3-form curvature vanish is canonically equivalent to
the 2-groupoid of Lie 2-algebra valued forms for which the 2-form curvature vanishes is canonically equivalent to
The equivalence is given by 2-dimensional parallel transport. A proof is in SchrWalII.
The following proposition asserts that the Lie 2-groupoid of Lie 2-algebra valued forms is the coefficient object for for differential nonabelian cohomology in degree 2, namely for connections on principal 2-bundles and in particular on gerbes.
(2-bundles with connection)
For $X$ a paracompact smooth manifold and $\{U_i \to X\}$ a good open cover the 2-groupoid, let $X \stackrel{\simeq}{\leftarrow} C(\{U_i\})$ be the corresponding Cech nerve smooth 2-groupoid. Then
is equivalent to the 2-groupoid of $G$-principal 2-bundles with 2-connection.
This is discussed and proven in SchrWalII for the case where the 2-form curvature is restricted to vanish. In this case the above can be written as
where $P_2(C(\{\U_i\}) \in 2LieGrpd$ is a resolution of the path 2-groupoid of $X$.
2-groupoid of Lie 2-algebra valued forms
The 2-groupoid of Lie 2-algebra valued forms described in definition 2.11 of
There are many possible conventions. The one reproduced above is supposed to describe the bidual opposite 2-category of the 2-groupoid as defined in that article, with the direction of 1- and 2-morphisms reversed.
See also
differential cohomology in an (∞,1)-topos – survey - connections on 2-bundles.