Definition 24.12.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A sheaf of differential graded $\mathcal{O}$-algebras or a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$ is a cochain complex $\mathcal{A}^\bullet$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps

$\mathcal{A}^ n \times \mathcal{A}^ m \to \mathcal{A}^{n + m},\quad (a, b) \longmapsto ab$

called the multiplication maps with the following properties

1. multiplication is associative,

2. there is a global section $1$ of $\mathcal{A}^0$ which is a two-sided identity for multiplication,

3. for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $a \in \mathcal{A}^ n(U)$, and $b \in \mathcal{A}^ m(U)$ we have

$\text{d}^{n + m}(ab) = \text{d}^ n(a)b + (-1)^ n a\text{d}^ m(b)$

We often denote such a structure $(\mathcal{A}, \text{d})$. A homomorphism of differential graded $\mathcal{O}$-algebras from $(\mathcal{A}, \text{d})$ to $(\mathcal{B}, \text{d})$ is a map $f : \mathcal{A}^\bullet \to \mathcal{B}^\bullet$ of complexes of $\mathcal{O}$-modules compatible with the multiplication maps.

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