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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