I am trying to learn about $\mathbb{Z}$-graded manifolds. It seems that the theory of $\mathbb{Z}$-graded manifolds has some complications that supermanifolds do not have, and there is fewer literature about $\mathbb{Z}$-graded manifolds than those of supermanifolds.
Could you recommend me some introductory references about $\mathbb{Z}$-graded manifolds? It will be great if it contains some concrete examples.
It seems that the main approach to $\mathbb{Z}$-graded manifolds is made by using the language of sheaves, but I wonder if there is any concrete approach. To add some context, in the book A. Rogers of supermanifolds, both concrete approach and algebro-geometric approach are introduced. The latter one is the one that uses sheaves. I am asking if there is an analogue of concrete approach for $\mathbb{Z}$-graded manifolds. If such a thing is known, please give me a reference.
In the talk Formality theorem for differential graded manifolds the speaker gives the following definition of $\mathbb{Z}$-graded manifolds. What does the notation $\widehat{\otimes}_k$ means? Also, is it a standard definition of $\mathbb{Z}$-graded manifolds?
Let $k=\mathbb{R}$ or $\mathbb{C}$ and $C^{\infty}_M$ is a sheaf of smooth $k$-valued functions on a smooth manifold $M$. Then a $\mathbb{Z}$-graded manifold $\mathcal{M}$ with body $M$ is a sheaf $\mathcal{R}$ of $\mathbb{Z}$-graded commutative $C^{\infty}_M$-algebras over $M$ such that $$\mathcal{R}\cong C^{\infty}_M(U)\widehat{\otimes}_k \widehat{S}(V^{\vee})$$ for sufficiently small open subsets $U$ of $M$ and some graded $\mathbb{Z}$-graded $k$-vector space $V$. Here $\widehat{S}(V^{\vee})$ denotes the graded algebra of formal power series on $V$.
