This is about the notion of representations upto homotopy of Lie algebroids. I am following the reference Representations up to homotopy of Lie algebroids by Camilo Arias Abad and Marius Crainic.
Let $M$ be a smooth manifold and $A\rightarrow M$ a Lie algebroid over $M$. The proposition $3.2$ in above mentioned paper says that a representation upto homotopy is given by the following data:
- a graded vector bundle $E\rightarrow M$ (I think they are considering $\mathbb{Z}$ grading $E=\oplus_{i \in \mathbb{Z}}E^i$),
- a degree $1$ operator $\partial :E\rightarrow E$ making $(E,\partial)$ a complex (with $\partial:E^i \rightarrow E^{i+1}$),
- an $A$-connection on $E\rightarrow M$ (a nice map $\nabla:\Gamma(M,A)\times \Gamma(M,E)\rightarrow \Gamma(M,E)$) that is compatible with the complex structure $\partial:E\rightarrow E$,
- and $\rm{End}(E)$-valued $2$-form $\omega_2$ of total degree $1$, i.e., $$\omega_2\in \Omega^2(A, \underline{\rm{End}}^{-1}(E))$$ satisfying the condition $\partial(\omega_2)+R_\nabla=0$, where $R_\nabla$ is the curvature of $\nabla$,
- for each $i>2$ and $\rm{End}(E)$-valued $i$-form $\omega_i$ of total degree $1$, i.e., $$\omega_i\in \Omega^i(A,\underline{\rm{End}}^{1-i}(E))$$ satisfying the condition $\partial(\omega_i)+d_\nabla(\omega_{i-1})+\omega_2\wedge\omega_{i-2}+\cdots+\omega_{i-2}\wedge\omega_2=0$.
The question is as I mentioned in the title: how should one think about representations upto homotopy?
Representation of a Lie algebroid $A\rightarrow M$ is a vector bundle $E\rightarrow M$, together with ''an action'' of $A\rightarrow M$ on $E\rightarrow M$. I used the term action to remember (and relate) the notion of representation of a Lie group/Lie algebra etc. Here, the "action" is the data of a map $$\nabla:\Gamma(M,A)\times \Gamma(M,E)\rightarrow \Gamma(M,E)$$ that is nice to some extent; precisely, we want $\nabla$ to be a flat $A$-connection. The curvature $R_\nabla$ would then be zero. This is what I understand about representations of Lie algebroids.
The punch line I heard multiple times is, representation upto homotopy of a Lie algebroid $A\rightarrow M$ is a graded vector bundle with an $A$-connection that is flat upto some correction.
So, the first three conditions are understandable. The third condition says, the curvature $R_\nabla$ may not be zero, but there is a $2$-form $\omega_2$ with $\partial(\omega_2)+R_\nabla=0$.
If the degrees of $\omega_2, \omega_{i-2}$ matches appropriately, then we would just have $\omega_2\wedge \omega_{i-2}+\omega_{i-2}\wedge\omega_2=0$. Assuming this is the case for each component in $4$th condition, we would have $\partial(\omega_i)+d_\nabla(\omega_{i-1})=0$. But, the degrees may not match that well to give us $\partial(\omega_i)+d_\nabla(\omega_{i-1})=0$. The condition $\partial(\omega_i)+d_\nabla(\omega_{i-1})=0$ is still fine. But, the condition $$\partial(\omega_i)+d_\nabla(\omega_{i-1})+\omega_2\wedge\omega_{i-2}+\cdots+\omega_{i-2}\wedge\omega_2=0$$ is too difficult to digest. How should one think about this condition?
