Example 22.25.5 (Graded category of graded objects). Let $\mathcal{B}$ be an additive category. Recall that we have defined the category $\text{Gr}(\mathcal{B})$ of graded objects of $\mathcal{B}$ in Homology, Definition 12.16.1. In this example, we will construct a graded category $\text{Gr}^{gr}(\mathcal{B})$ over $R = \mathbf{Z}$ whose associated category $\text{Gr}^{gr}(\mathcal{B})^0$ recovers $\text{Gr}(\mathcal{B})$. As objects of $\text{Gr}^{gr}(\mathcal{B})$ we take graded objects of $\mathcal{B}$. Then, given graded objects $A = (A^ i)$ and $B = (B^ i)$ of $\mathcal{B}$ we set

where the graded piece of degree $n$ is the abelian group of homogeneous maps of degree $n$ from $A$ to $B$. Explicitly we have

(observe reversal of indices and observe that we have a product here and not a direct sum). In other words, a degree $n$ morphism $f$ from $A$ to $B$ can be seen as a system $f = (f_{p, q})$ where $p, q \in \mathbf{Z}$, $p + q = n$ with $f_{p, q} : A^{-q} \to B^ p$ a morphism of $\mathcal{B}$. Given graded objects $A$, $B$, $C$ of $\mathcal{B}$ composition of morphisms in $\text{Gr}^{gr}(\mathcal{B})$ is defined via the maps

by simple composition $(g, f) \mapsto g \circ f$ of homogeneous maps of graded objects. In terms of components we have

where $q$ is such that $p + q = m$ and $-q + r = n$.

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