Someone has mentioned to me that generalizations of co-chain complexes and their cohomology have been studied, where instead of $d^2 = 0$ we have something like $d^2 \alpha = q \alpha $, which is known as "curvature".
Where can I find out more about this, and how does cohomology generalize, if at all, in this setting?
If someone could provide a reference for this, or point me in the general direction of one, I would be very grateful.
Related: curved $A_\infty$-algebras $\mathcal A:=(A_\bullet, d)$ are endowed with a pseudo differential $d_1:A_n\rightarrow A_{n+1}$ s.t.
$$d_1\circ d_1(a)\pm d_2(a,d_0(1))\pm d_2(d_0(1),a)=0$$
for all $a\in \mathcal A$, denoting by $d_0(1)\in A_2$ the curvature of the algebra ( $1$ is the unit) and with $d_2$ the binary product in $\mathcal A$. Clearly, if the curvature vanishes, then $d_1$ is a differential.
Such generalizations of flat $A_\infty$ algebras are used in homological algebra and have connections with theoretical physics (through Landau Ginzburg models, for example). To my knowledge no sound definition of cohomology of these algebras has been set up, and the derived category machinery is missing.
Check the works of Keller, Positselski for more homological background.