We make the following definition.
Definition 12.16.1. Let $\mathcal{A}$ be an additive category. The category of graded objects of $\mathcal{A}$, denoted $\text{Gr}(\mathcal{A})$, is the category with
objects $A = (A^ i)$ are families of objects $A^ i$, $i \in \mathbf{Z}$ of objects of $\mathcal{A}$, and
morphisms $f : A = (A^ i) \to B = (B^ i)$ are families of morphisms $f^ i : A^ i \to B^ i$ of $\mathcal{A}$.
If $\mathcal{A}$ has countable direct sums, then we can associate to an object $A = (A^ i)$ of $\text{Gr}(\mathcal{A})$ the object
and set $k^ iA = A^ i$. In this case $\text{Gr}(\mathcal{A})$ is equivalent to the category of pairs $(A, k)$ consisting of an object $A$ of $\mathcal{A}$ and a direct sum decomposition
by direct summands indexed by $\mathbf{Z}$ and a morphism $(A, k) \to (B, k)$ of such objects is given by a morphism $\varphi : A \to B$ of $\mathcal{A}$ such that $\varphi (k^ iA) \subset k^ iB$ for all $i \in \mathbf{Z}$. Whenever our additive category $\mathcal{A}$ has countable direct sums we will use this equivalence without further mention.
However, with our definitions an additive or abelian category does not necessarily have all (countable) direct sums. In this case our definition still makes sense. For example, if $\mathcal{A} = \text{Vect}_ k$ is the category of finite dimensional vector spaces over a field $k$, then $\text{Gr}(\text{Vect}_ k)$ is the category of vector spaces with a given gradation all of whose graded pieces are finite dimensional, and not the category of finite dimensional vector spaces with a given graduation.
Lemma 12.16.2. Let $\mathcal{A}$ be an abelian category. The category of graded objects $\text{Gr}(\mathcal{A})$ is abelian.
Proof. Let $f : A = (A^ i) \to B = (B^ i)$ be a morphism of graded objects of $\mathcal{A}$ given by morphisms $f^ i : A^ i \to B^ i$ of $\mathcal{A}$. Then we have $\mathop{\mathrm{Ker}}(f) = (\mathop{\mathrm{Ker}}(f^ i))$ and $\mathop{\mathrm{Coker}}(f) = (\mathop{\mathrm{Coker}}(f^ i))$ in the category $\text{Gr}(\mathcal{A})$. Since we have $\mathop{\mathrm{Im}}= \mathop{\mathrm{Coim}}$ in $\mathcal{A}$ we see the same thing holds in $\text{Gr}(\mathcal{A})$. $\square$
Remark 12.16.3 (Warning). There are abelian categories $\mathcal{A}$ having countable direct sums but where countable direct sums are not exact. An example is the opposite of the category of abelian sheaves on $\mathbf{R}$. Namely, the category of abelian sheaves on $\mathbf{R}$ has countable products, but countable products are not exact. For such a category the functor $\text{Gr}(\mathcal{A}) \to \mathcal{A}$, $(A^ i) \mapsto \bigoplus A^ i$ described above is not exact. It is still true that $\text{Gr}(\mathcal{A})$ is equivalent to the category of graded objects $(A, k)$ of $\mathcal{A}$, but the kernel in the category of graded objects of a map $\varphi : (A, k) \to (B, k)$ is not equal to $\mathop{\mathrm{Ker}}(\varphi )$ endowed with a direct sum decomposition, but rather it is the direct sum of the kernels of the maps $k^ iA \to k^ iB$.
Definition 12.16.4. Let $\mathcal{A}$ be an additive category. If $A = (A^ i)$ is a graded object, then the $k$th shift $A[k]$ is the graded object with $A[k]^ i = A^{k + i}$.
If $A$ and $B$ are graded objects of $\mathcal{A}$, then we have
and an element of this group is sometimes called a map of graded objects homogeneous of degree $k$.
Given any set $G$ we can define $G$-graded objects of $\mathcal{A}$ as the category whose objects are $A = (A^ g)_{g \in G}$ families of objects parametrized by elements of $G$. Morphisms $f : A \to B$ are defined as families of maps $f^ g : A^ g \to B^ g$ where $g$ runs over the elements of $G$. If $G$ is an abelian group, then we can (unambiguously) define shift functors $[g]$ on the category of $G$-graded objects by the rule $(A[g])^{g_0} = A^{g + g_0}$. A particular case of this type of construction is when $G = \mathbf{Z} \times \mathbf{Z}$. In this case the objects of the category are called bigraded objects of $\mathcal{A}$. The $(p, q)$ component of a bigraded object $A$ is usually denoted $A^{p, q}$. For $(a, b) \in \mathbf{Z} \times \mathbf{Z}$ we write $A[a, b]$ in stead of $A[(a, b)]$. A morphism $A \to A[a, b]$ is sometimes called a map of bidegree $(a, b)$.
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