The Bianchi identity is a differential equation satisfied by curvature data.
It can be thought of as generalizing the equation $d (d A) = 0$ for a real-valued 1-form $A$ to higher degree and nonabelian forms.
Generally it applies to the curvature of ∞-Lie algebroid valued differential forms.
Let $U$ be a smooth manifold.
For $A \in \Omega^1(U)$ a differential 1-form, its curvature 2-form is the de Rham differential $F_a = d A$. The Bianchi identity in this case is the equation
More generally, for $\mathfrak{g}$ an arbitrary Lie algebra and $A \in \Omega^1(U,\mathfrak{g})$ a Lie-algebra valued 1-form, its curvature is the 2-form $F_A = d A + [A \wedge A]$. The Bianchi identity in this case is the equation
satisfied by these curvature 2-forms.
We may reformulate the above identities as follows.
For $\mathfrak{g}$ a Lie algebra we have naturally associated two dg-algebras: the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ and the Weil algebra $W(\mathfrak{g})$.
The dg-algebra morphisms
are precisely in bijection with Lie-algebra valued 1-forms as follows: the Weil algebra is of the form
with one copy of $\mathfrak{g}^*$ in degree 1, the other in degree 2. By the free nature of the Weil algebra, dg-algebra morphisms $\Omega^\bullet(U) \leftarrow W(\mathfrak{g})$ are in bijection to their underlying morphisms of vector spaces of generators
This identifies the 1-form $A \in \Omega^1(U,\mathfrak{g})$. This extends uniquely to a morphism of dg-algebras and thereby fixes the image of the shifted generators
The Bianchi identity is precisely the statement that these linear maps, extended to morphisms of graded algebra, are compatible with the differentials and hence do constitute dg-algebra morphisms.
Concretely, if $\{t^a\}$ is a dual basis for $\mathfrak{g}^*$ and $\{r^a\}$ the corresponding dual basis for $\mathfrak{g}^*[1]$ and $\{C^a{}_{b c}\}$ the structure constants of the Lie bracket $[-,-]$ on $\mathfrak{g}$, then the differential $d_{W(\mathfrak{g})}$ on the Weil algebra is defined on generators by
and
The image of $t^a$ under $\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,F_A)$ is the component $A^a$. The image of $r^a$ is therefore, by respect for the differential on $t^a$
Respect for the differential on $r^a$ then implies
This is the Bianchi identity.
Let now $\mathfrak{g}$ be an arbitrary ∞-Lie-algebra and $W(\mathfrak{g})$ its Weil algebra. Then a collection of ∞-Lie algebra valued differential forms is a dg-algebra morphism
It curvature is the composite of morphism of graded vector space
Since $A$ is a homomorphism of dg-algebras, this satisfies
This identity is the Bianchi identity for $\infty$-Lie algebra valued forms.
Bianchi identity
The Bianchi identity for ∞-Lie algebroid valued differential forms is discussed in