This is from the paper Representations up to homotopy of Lie algebroids by Camilo Arias Abad and Marius Crainic

Let $M$ be a smooth manifold.

Let $E\rightarrow M$ be a vector bundle. A connection on the vector bundle $E\rightarrow M$ is a map $$\nabla:\Gamma(M,TM)\times \Gamma(M,E)\rightarrow \Gamma(M,E)$$ satisfying certain conditions.

Let $A\rightarrow M$ be a Lie algebroid on $M$. An $A$-connection on the vector bundle $E\rightarrow M$ is a map $$\nabla:\Gamma(M,A)\times \Gamma(M,E)\rightarrow\Gamma(M,E)$$ satisfying same conditions as mentioned before :D

This notion seems to be introduced in the above paper, please correct me if I am wrong.

After Definition $2.9$ in the above paper, the authors mention about the notion of $A$-connection on the (adjoint) complex (of vector bundles). But, the authors do not even declare the meaning of the notion of connection on chain/cochain complex of vector bundles.

Can someone suggest some reference where I can find a meaning to this notion?

I can make a guess but I am sure the exact notion is more than what I can guess.

Consider the adjoint complex $\rho:A\rightarrow TM$ (which, for me is just a nice morphism of vector bundles).

An $A$-connection on the complex $\rho:A\rightarrow TM$ should be just a pair $(\nabla_A,\nabla_{TM})$ where $\nabla_A$ is an $A$-connection on the vector bundle $A\rightarrow M$, and $\nabla_{TM}$ is an $A$-connection on the vector bundle $TM\rightarrow M$ such that, they are connected with each other with the help of the morphism $\rho:A\rightarrow TM$.

So, what exactly does it mean to say connection on ($2$-term) complex of vector bundles?

This is from the paper Representations up to homotopy of Lie algebroids by Camilo Arias Abad and Marius Crainic

Let $M$ be a smooth manifold.

Let $E\rightarrow M$ be a vector bundle. A connection on the vector bundle $E\rightarrow M$ is a map $$\nabla:\Gamma(M,TM)\times \Gamma(M,E)\rightarrow \Gamma(M,E)$$ satisfying certain conditions.

Let $A\rightarrow M$ be a Lie algebroid on $M$. An $A$-connection on the vector bundle $E\rightarrow M$ is a map $$\nabla:\Gamma(M,A)\times \Gamma(M,E)\rightarrow\Gamma(M,E)$$ satisfying the same conditions as mentioned before. :D

This notion seems to be introduced in the above paper, please correct me if I am wrong.

After Definition $2.9$ in the above paper, the authors mention the notion of an $A$-connection on the (adjoint) complex (of vector bundles). But, the authors do not even declare the meaning of the notion of connection on a chain/cochain complex of vector bundles.

Can someone suggest some reference where I can find a meaning to this notion?

I can make a guess but I am sure the exact notion is more than what I can guess.

Consider the adjoint complex $\rho:A\rightarrow TM$ (which, for me is just a nice morphism of vector bundles).

An $A$-connection on the complex $\rho:A\rightarrow TM$ should be just a pair $(\nabla_A,\nabla_{TM})$ where $\nabla_A$ is an $A$-connection on the vector bundle $A\rightarrow M$, and $\nabla_{TM}$ is an $A$-connection on the vector bundle $TM\rightarrow M$ such that, they are connected with each other with the help of the morphism $\rho:A\rightarrow TM$.

So, what exactly does it mean to refer to a connection on a ($2$-term) complex of vector bundles?