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Definition 22.5.1. Let $(A, \text{d})$ be a differential graded algebra. Let $f, g : M \to N$ be homomorphisms of differential graded $A$-modules. A homotopy between $f$ and $g$ is an $A$-module map $h : M \to N$ such that

  1. $h(M^ n) \subset N^{n - 1}$ for all $n$, and

  2. $f(x) - g(x) = \text{d}_ N(h(x)) + h(\text{d}_ M(x))$ for all $x \in M$.

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

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