Note however that in the more general case where ρ does not have constant rank we cannot easily define the representations g(A) and ν(A).
A representation up to homotopy as introduced above is equivalent to the following data The correspondence is characterized as A homomorphism between representations up to homotopy (E,DE) and (F,DF) of the same Lie algebroid A is a degree 0 map Φ:Ω(A,E) → Ω(A,F) that commutes with the differentials, i.e. An isomorphism is now an invertible homomorphism.
In the sense of the above decomposition of D into a cochain map ∂, a connection ∇, and higher homotopies, we can also decompose the Φ as Φ0 + Φ1 + ... with and then the compatibility condition reads Examples are usual representations of Lie algebroids or more specifically Lie algebras, i.e. modules.
together with a connection ∇ on its vector bundle we can define two associated A-connections as follows[3] Moreover, we can introduce the mixed curvature as This curvature measures the compatibility of the Lie bracket with the connection and is one of the two conditions of A together with TM forming a matched pair of Lie algebroids.
The first observation is that this term decorated with the anchor map ρ, accordingly, expresses the curvature of both connections ∇bas.