nLab connection for a differential graded algebra

Given a semi-free differential graded algebra $\Omega^\bullet A$ over $k$-algebra $A$, a connection in $A$-module $M$ is a $k$-linear map

$$\nabla : M\otimes_A\Omega^\bullet A \to M\otimes_A\Omega^{\bullet+1} A$$

such that for any homogenous element $\omega\in\Omega^k A$ and an element $\chi\in\Omega A$

$$\nabla (\omega\chi) = \nabla(\omega)\chi + (-1)^k\omega\nabla(\chi)$$

The curvature $F_\nabla$ of the connection $\nabla$ is the restriction of the connection squared to $M$:

$$\nabla\circ\nabla|_M : M\to M\otimes_A\Omega^2 A.$$

A connection is flat (or integrable) iff its curvature vanishes.

See also connection in noncommutative geometry as some versions are close to this approach. A past query on history of the notion of connection for a dga is archived here.