The Stacks project

Definition 24.21.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $f, g : \mathcal{M} \to \mathcal{N}$ be homomorphisms of differential graded $\mathcal{A}$-modules. A homotopy between $f$ and $g$ is a graded $\mathcal{A}$-module map $h : \mathcal{M} \to \mathcal{N}$ homogeneous of degree $-1$ such that

\[ f - g = \text{d}_\mathcal {N} \circ h + h \circ \text{d}_\mathcal {M} \]

If a homotopy exists, then we say $f$ and $g$ are homotopic.

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