nLab 2-groupoid of Lie 2-algebra valued forms

Idea

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.

Definition

For strict Lie 2-algebras

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

$$[\mathbf{B}G] = (G_2 \stackrel{\delta}{\to} G_1 \stackrel{\to}{\to} *) \,.$$

Write $[\mathfrak{g}_2 \stackrel{\delta_*}{\to} \mathfrak{g}_1]$ for the corresponding differential crossed module with action $\alpha_* : \mathfrak{g}_1 \to der(\mathfrak{g}_2)$

Definition

The 2-groupoid of Lie 2-algebra valued forms is defined to be the 2-stack

$$\bar \mathbf{B}G : CartSp{}^{op} \to 2Grpd$$

which assigns to $U \in CartSp$ the following 2-groupoid:

• An object is a pair

  $$A \in \Omega^1(U,\mathfrak{g}_1)\,, \;\;\; B \in \Omega^2(U,\mathfrak{g}_2) \,.$$

• A 1-morphism $(g,a) : (A,B) \to (A',B')$ is a pair

  $$g \in C^\infty(U,G_1)\,,\;\;\; a \in \Omega^1(U,\mathfrak{g}_2)$$

  such that

  $$A' = Ad_{g^{-1}}\left( A + \delta_* a \right) + g^{-1} d g$$

  and

  $$B' = \alpha_{g^{-1}}( B + d a + [a \wedge a] + \alpha_*(A \wedge a) ) \,.$$

  The composite of two 1-morphisms

  $$(A,B) \stackrel{(g_1,a_1)}{\to} (A',B') \stackrel{(g_2,a_2)}{\to} (A'', B'')$$

  is given by the pair

  $$(g_1 g_2, a_1 + (\alpha_{g_2})_* a_2) \,.$$

• a 2-morphism $f : (g,a) \Rightarrow (g', a'):(A,B)\to (A',B')$ is a function

  $$f \in C^\infty(U,G_2)$$

  such that

  $$g' = \delta(f)^{-1} \cdot g$$

  and

  $$a' = Ad_{f^{-1}} \left(a + (r_f^{-1} \circ \alpha_f)_*(A)\right) + f^{-1} d f$$

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 $\mathbf{B}G)$.

For general Lie 2-algebras

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

$$\{t^a\} \,, \;\;\; \{b^i\}$$

be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general Chevalley-Eilenberg algebra

$$CE(\mathfrak{g}) \in cdgAlg_\mathbb{R}$$

with these generators.

We thus have

$$d_{CE(\mathfrak{g})} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c - r^a{}_i b^i$$

$$d_{CE(\mathfrak{g})} b^i = -\alpha^i_{a j} t^a \wedge b^j - r_{a b c} t^a \wedge t^b \wedge t^c \,,$$

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

$$(d_{CE(\mathfrak{g})})^2 = 0 \,.$$

Over a test space $U$ a $\mathfrak{g}$-valued form datum is a morphism

$$\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,B)$$

from the Weil algebra $W(\mathfrak{g})$.

This is given by a 1-form

$$A \in \Omega^1(U, \mathfrak{g}_0)$$

and a 2-form

$$B \in \Omega^2(U, \mathfrak{g}_1) \,.$$

The curvature of this is $(\beta, H)$, where the 2-form component (“fake curvature”) is

$$\beta^a = d_{dR} A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c + r^a{}_i B^i$$

and whose 3-form component is

$$H^i = d_{dR} B^i + \alpha^i{}_{a j} A^a \wedge B^j + t_{a b c} A^a \wedge A^b \wedge A^c \,.$$

Properties

Proposition

(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

$$\delta_* B - d A + [A \wedge A] = 0$$

and the 3-form curvature

$$d B + [A \wedge B] = 0$$

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.

Proposition

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

  $$Hom_{2Grpd(Diffeol)}(\Pi_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd \,;$$

• the 2-groupoid of Lie 2-algebra valued forms for which the 2-form curvature vanishes is canonically equivalent to

  $$Hom_{2Grpd(Diffeol)}(P_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd \,;$$

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.

Proposition

(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

$$Hom_{2Grpd(Diffeol)}( C(\{U_i\}), \bar \mathbf{B}G)$$

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

$$Hom( C(\{U_i\}), Hom(P_2(-), \mathbf{B}G)) \simeq Hom(P_2(C(\{\U_i\})), \mathbf{B}G) \,,$$

where $P_2(C(\{\U_i\}) \in 2LieGrpd$ is a resolution of the path 2-groupoid of $X$.

Related concepts

• groupoid of Lie-algebra valued forms

• 2-groupoid of Lie 2-algebra valued forms

  • nonabelian Stokes theorem

• 3-groupoid of Lie 3-algebra valued forms

• ∞-groupoid of ∞-Lie-algebra valued forms

References

The 2-groupoid of Lie 2-algebra valued forms described in definition 2.11 of

• Schreiber, Waldorf, Smooth functors versus differential forms (web).

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.