Example 22.26.6 (Differential graded category of complexes). Let \mathcal{B} be an additive category. We will construct a differential graded category \text{Comp}^{dg}(\mathcal{B}) over R = \mathbf{Z} whose associated category of complexes is \text{Comp}(\mathcal{B}) and whose associated homotopy category is K(\mathcal{B}). As objects of \text{Comp}^{dg}(\mathcal{B}) we take complexes of \mathcal{B}. Given complexes A^\bullet and B^\bullet of \mathcal{B}, we sometimes also denote A^\bullet and B^\bullet the corresponding graded objects of \mathcal{B} (i.e., forget about the differential). Using this abuse of notation, we set
as a graded \mathbf{Z}-module with notation and definitions as in Example 22.25.5. In other words, the nth graded piece is the abelian group of homogeneous morphism of degree n of graded objects
Observe reversal of indices and observe we have a direct product and not a direct sum. For an element f \in \mathop{\mathrm{Hom}}\nolimits ^ n(A^\bullet , B^\bullet ) of degree n we set
The sign is exactly as in More on Algebra, Section 15.73. To make sense of this we think of \text{d}_ B and \text{d}_ A as maps of graded objects of \mathcal{B} homogeneous of degree 1 and we use composition in the category \text{Gr}^{gr}(\mathcal{B}) on the right hand side. In terms of components, if f = (f_{p, q}) with f_{p, q} : A^{-q} \to B^ p we have
Note that the first term of this expression is in \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(A^{-q}, B^{p + 1}) and the second term is in \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(A^{-q - 1}, B^ p). The reader checks that
\text{d} has square zero,
an element f in \mathop{\mathrm{Hom}}\nolimits ^ n(A^\bullet , B^\bullet ) has \text{d}(f) = 0 if and only if the morphism f : A^\bullet \to B^\bullet [n] of graded objects of \mathcal{B} is actually a map of complexes,
in particular, the category of complexes of \text{Comp}^{dg}(\mathcal{B}) is equal to \text{Comp}(\mathcal{B}),
the morphism of complexes defined by f as in (2) is homotopy equivalent to zero if and only if f = \text{d}(g) for some g \in \mathop{\mathrm{Hom}}\nolimits ^{n - 1}(A^\bullet , B^\bullet ).
in particular, we obtain a canonical isomorphism
\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{B})}(A^\bullet , B^\bullet ) \longrightarrow H^0(\mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{B})}(A^\bullet , B^\bullet ))and the homotopy category of \text{Comp}^{dg}(\mathcal{B}) is equal to K(\mathcal{B}).
Given complexes A^\bullet , B^\bullet , C^\bullet we define composition
by composition (g, f) \mapsto g \circ f in the graded category \text{Gr}^{gr}(\mathcal{B}), see Example 22.25.5. This defines a map of differential graded modules
as required in Definition 22.26.1 because
as desired.
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