The Stacks project

Definition 24.28.2. Derived tensor product and derived pullback.

1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. The functor $D(\mathcal{A}, \text{d}) \to D(\mathcal{B}, \text{d})$ constructed in Lemma 24.28.1 is called the derived tensor product and denoted $- \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}$.

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. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal {C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal {D}$-algebra. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal {D}$-algebras. The functor $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}, \text{d})$ constructed in Lemma 24.28.1 is called derived pullback and denote $Lf^*$.