The Stacks project

Remark 24.12.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi.

1. Let $(\mathcal{A}, \text{d})$ be a differential graded $\mathcal{O}_\mathcal {C}$-algebra. The pushforward will be the differential graded $\mathcal{O}_\mathcal {D}$-algebra $(f_*\mathcal{A}, \text{d})$ where $f_*\mathcal{A}$ is as in Remark 24.3.2 and $\text{d} = f_*\text{d}$ as maps $f_*\mathcal{A}^ n \to f_*\mathcal{A}^{n + 1}$. We omit the verification that the Leibniz rule is satisfied.

2. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal {D}$-algebra. The pullback will be the differential graded $\mathcal{O}_\mathcal {C}$-algebra $(f^*\mathcal{B}, \text{d})$ where $f^*\mathcal{B}$ is as in Remark 24.3.2 and $\text{d} = f^*\text{d}$ as maps $f^*\mathcal{B}^ n \to f^*\mathcal{B}^{n + 1}$. We omit the verification that the Leibniz rule is satisfied.

3. The set of homomorphisms $f^*\mathcal{B} \to \mathcal{A}$ of differential graded $\mathcal{O}_\mathcal {C}$-algebras is in $1$-to-$1$ correspondence with the set of homomorphisms $\mathcal{B} \to f_*\mathcal{A}$ of differential graded $\mathcal{O}_\mathcal {D}$-algebras.

Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$ on sheaves of modules.