Skew differential graded algebra

Question

A sigma, or skew, derivation is a natural generalisation of the notion of derivation depending on an algebra automorphism $\sigma$ which when equal to $id = \sigma$ reduces to the usual notion of a derivation. For a precise definition see here

https://planetmath.org/SigmaDerivation

Does there exist a notion of skew differential graded algebra in the literature? If so where do these objects arise?

EDIT: To confirm I am asking if there exists a graded analogue of skew derivation algebra. So an $\mathbb{N}_0$-graded algebra $A = \bigoplus_{k \in \mathbb{N}_)0} A_k$, together with a degree $1$ map $d$ satisfying $d^2 = 0$, and a skew analogue of the graded Leibniz rule: $$ d(a \wedge b) = da \wedge \sigma(b) + (-1)^k \sigma(a)db, ~ a \in A_k $$

Answer 1

This edited version of the "skew Leibniz rule" has appeared in geometry: if $\varphi: N\to M$ is a map of (super) manifolds, a section $X$ of the pullback bundle $\varphi^\star TM$ is a linear map $C^\infty(M)\to C^\infty(N)$ satisfying precisely your skew-leibniz rule: $$ X(fg)=X(f)\varphi^\star g +(-1)^{\deg X\deg f} (\varphi^\star f) X(g)$$

(See J. Nestruev, Smooth manifolds and observables, paragraph 9.47.)

So if you put $\mathbb N_0$-gradings on your structure sheaves (what is $\mathbb N_0$? Positive integers?) and pick an $X$ of degree $+1$ which squares to zero you seem to arrive at your setup. (Identifying $\varphi^\star$ with $\sigma$ and $C^\infty(M)$ with $A$.)

On the algebraic side there is the (comparatively more obscure) notion of $({\bf l},{\bf r})$-coderivation of Berglund (Definition 3.2) which should be dual to your proposal in the case ${\bf l}={\bf r}=\sigma$.