24.21 The homotopy category
This section is the analogue of Differential Graded Algebra, Section 22.5.
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.
In the situation of the definition, if we have maps $a : \mathcal{K} \to \mathcal{M}$ and $c : \mathcal{N} \to \mathcal{L}$ then we see that
\[ \begin{matrix} h\text{ is a homotopy}
\\ \text{ between }f\text{ and } g
\end{matrix} \quad \Rightarrow \quad \begin{matrix} c \circ h \circ a\text{ is a homotopy}
\\ \text{between } c \circ f \circ a\text{ and } c\circ g \circ a
\end{matrix} \]
Thus we can define composition of homotopy classes of morphisms in $\textit{Mod}(\mathcal{A}, \text{d})$.
Definition 24.21.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The homotopy category, denoted $K(\textit{Mod}(\mathcal{A}, \text{d}))$, is the category whose objects are the objects of $\textit{Mod}(\mathcal{A}, \text{d})$ and whose morphisms are homotopy classes of homomorphisms of differential graded $\mathcal{A}$-modules.
The notation $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is not standard but at least is consistent with the use of $K(-)$ in other places of the Stacks project.
In Differential Graded Algebra, Definition 22.26.3 we have defined what we mean by the category of complexes $\text{Comp}(\mathcal{S})$ and the homotopy category $K(\mathcal{S})$ of a differential graded category $\mathcal{S}$. Applying this to the differential graded category $\textit{Mod}^{dg}(\mathcal{A}, \text{d})$ we obtain
\[ \textit{Mod}(\mathcal{A}, \text{d}) = \text{Comp}(\textit{Mod}^{dg}(\mathcal{A}, \text{d})) \]
(see discussion in Section 24.14) and we obtain
\[ K(\textit{Mod}(\mathcal{A}, \text{d})) = K(\textit{Mod}^{dg}(\mathcal{A}, \text{d})) \]
To see that this last equality is true, note that we have the equality
\[ \text{d}_{\mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})} (\mathcal{M}, \mathcal{N})}(h) = \text{d}_\mathcal {N} \circ h + h \circ \text{d}_\mathcal {M} \]
when $h$ is as in Definition 24.21.1. We omit the details.
Lemma 24.21.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The homotopy category $K(\textit{Mod}(\mathcal{A}, \text{d}))$ has direct sums and products.
Proof.
Omitted. Hint: Just use the direct sums and products as in Lemma 24.13.2. This works because we saw that these functors commute with the forgetful functor to the category of graded $\mathcal{A}$-modules and because $\prod $ and $\bigoplus $ are exact functors on the category of families of abelian groups.
$\square$
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