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

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

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

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

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