Just some definitions.

Definition 22.25.1. Let $R$ be a ring. A graded category $\mathcal{A}$ over $R$ is a category where every morphism set is given the structure of a graded $R$-module and where for $x, y, z \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ composition is $R$-bilinear and induces a homomorphism

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(y, z) \otimes _ R \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, z)$

of graded $R$-modules (i.e., preserving degrees).

In this situation we denote $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}^ i(x, y)$ the degree $i$ part of the graded object $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y)$, so that

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) = \bigoplus \nolimits _{i \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}^ i(x, y)$

is the direct sum decomposition into graded parts.

Definition 22.25.2. Let $R$ be a ring. A functor of graded categories over $R$, or a graded functor is a functor $F : \mathcal{A} \to \mathcal{B}$ where for all objects $x, y$ of $\mathcal{A}$ the map $F : \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(F(x), F(y))$ is a homomorphism of graded $R$-modules.

Given a graded category we are often interested in the corresponding “usual” category of maps of degree $0$. Here is a formal definition.

Definition 22.25.3. Let $R$ be a ring. Let $\mathcal{A}$ be a graded category over $R$. We let $\mathcal{A}^0$ be the category with the same objects as $\mathcal{A}$ and with

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{A}^0}(x, y) = \mathop{\mathrm{Hom}}\nolimits ^0_\mathcal {A}(x, y)$

the degree $0$ graded piece of the graded module of morphisms of $\mathcal{A}$.

Definition 22.25.4. Let $R$ be a ring. Let $\mathcal{A}$ be a graded category over $R$. A direct sum $(x, y, z, i, j, p, q)$ in $\mathcal{A}$ (notation as in Homology, Remark 12.3.6) is a graded direct sum if $i, j, p, q$ are homogeneous of degree $0$.

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

$\mathop{\mathrm{Hom}}\nolimits _{\text{Gr}^{gr}(\mathcal{B})}(A, B) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits ^ n(A, B)$

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

$\mathop{\mathrm{Hom}}\nolimits ^ n(A, B) = \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(A^{-q}, B^ p)$

(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

$\mathop{\mathrm{Hom}}\nolimits ^ m(B, C) \times \mathop{\mathrm{Hom}}\nolimits ^ n(A, B) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^{n + m}(A, C)$

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

$(g \circ f)_{p, r} = g_{p, q} \circ f_{-q, r}$

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

Example 22.25.6 (Graded category of graded modules). Let $A$ be a $\mathbf{Z}$-graded algebra over a ring $R$. We will construct a graded category $\text{Mod}^{gr}_ A$ over $R$ whose associated category $(\text{Mod}^{gr}_ A)^0$ is the category of graded $A$-modules. As objects of $\text{Mod}^{gr}_ A$ we take right graded $A$-modules (see Section 22.14). Given graded $A$-modules $L$ and $M$ we set

$\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{gr}_ A}(L, M) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits ^ n(L, M)$

where $\mathop{\mathrm{Hom}}\nolimits ^ n(L, M)$ is the set of right $A$-module maps $L \to M$ which are homogeneous of degree $n$, i.e., $f(L^ i) \subset M^{i + n}$ for all $i \in \mathbf{Z}$. In terms of components, we have that

$\mathop{\mathrm{Hom}}\nolimits ^ n(L, M) \subset \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(L^{-q}, M^ p)$

(observe reversal of indices) is the subset consisting of those $f = (f_{p, q})$ such that

$f_{p, q}(m a) = f_{p - i, q + i}(m)a$

for $a \in A^ i$ and $m \in L^{-q - i}$. For graded $A$-modules $K$, $L$, $M$ we define composition in $\text{Mod}^{gr}_ A$ via the maps

$\mathop{\mathrm{Hom}}\nolimits ^ m(L, M) \times \mathop{\mathrm{Hom}}\nolimits ^ n(K, L) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^{n + m}(K, M)$

by simple composition of right $A$-module maps: $(g, f) \mapsto g \circ f$.

Remark 22.25.7. Let $R$ be a ring. Let $\mathcal{D}$ be an $R$-linear category endowed with a collection of $R$-linear functors $[n] : \mathcal{D} \to \mathcal{D}$, $x \mapsto x[n]$ indexed by $n \in \mathbf{Z}$ such that $[n] \circ [m] = [n + m]$ and $[0] = \text{id}_\mathcal {D}$ (equality as functors). This allows us to construct a graded category $\mathcal{D}^{gr}$ over $R$ with the same objects of $\mathcal{D}$ setting

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}^{gr}}(x, y) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(x, y[n])$

for $x, y$ in $\mathcal{D}$. Observe that $(\mathcal{D}^{gr})^0 = \mathcal{D}$ (see Definition 22.25.3). Moreover, the graded category $\mathcal{D}^{gr}$ inherits $R$-linear graded functors $[n]$ satisfying $[n] \circ [m] = [n + m]$ and $[0] = \text{id}_{\mathcal{D}^{gr}}$ with the property that

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}^{gr}}(x, y[n]) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}^{gr}}(x, y)[n]$

as graded $R$-modules compatible with composition of morphisms.

Conversely, suppose given a graded category $\mathcal{A}$ over $R$ endowed with a collection of $R$-linear graded functors $[n]$ satisfying $[n] \circ [m] = [n + m]$ and $[0] = \text{id}_\mathcal {A}$ which are moreover equipped with isomorphisms

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y[n]) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y)[n]$

as graded $R$-modules compatible with composition of morphisms. Then the reader easily shows that $\mathcal{A} = (\mathcal{A}^0)^{gr}$.

Here are two examples of the relationship $\mathcal{D} \leftrightarrow \mathcal{A}$ we established above:

1. Let $\mathcal{B}$ be an additive category. If $\mathcal{D} = \text{Gr}(\mathcal{B})$, then $\mathcal{A} = \text{Gr}^{gr}(\mathcal{B})$ as in Example 22.25.5.

2. If $A$ is a graded ring and $\mathcal{D} = \text{Mod}_ A$ is the category of graded right $A$-modules, then $\mathcal{A} = \text{Mod}^{gr}_ A$, see Example 22.25.6.

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