The definition of graded algebra had a growing interest in algebra and mathematical physics (see $[GTC]$), I see that this topic has an elegant and simple categorical generalization, but I have not found in the literature.

**My question is whether there is such a categorical description in the literature, with further development than my observations:**

Let $\mathcal{C}$ a $R$-additive monoidal symmetrical category ($R$ is a commutative ring), let $AbM$ the category of commutative (additive) monoids.

I define the category $Gr.\mathcal{C}$ with objects as $(A, G)$ where $A\in AbM$ with a simple functor $G: |A|\to \mathcal{C}$ where $|A|$ is the support set of $A$, i.e. a family of objects $G_a,\ a\in A$ of $\mathcal{C}$, in simple words we call $G$ a $A$-graded object.

With morphisms as $(f, \phi): (A, G)\to (B, H)$ where $f: A\to B$ a monoids morphism, and $\phi: G\Rightarrow f^\ast H$ where $(f^\ast H)(a)= H_{f(a)}$ i.e. a family of morphisms $\phi_a: G_a \to H_{f(a)}\ a\in A$.

Naturally the natural functor $Gr.\mathcal{C}\to AbM$ is a strict-fibration with cartesian morphisms as $(f, 1)$, if $\mathcal{C}$ has sums then this functor is also a strict-cofibration, with $(f_\ast G)(b)= \oplus_{f(a)=b} G_a$.

Given $A$-graded abject $(A, G), (A, H)$ define the monoidal product $G\otimes H$ as the $A\times A$-graded object: $\otimes \circ (G\times H): |A|\times |A|\to\mathcal{C}\times \mathcal{C}\to\mathcal{C}$ (or eventually as the $A$-graded $(+)_\ast(G\otimes H)$ where $+: A\times A\to A$).

Naturally $Gr.\mathcal{C}$ become a symmetrical monoidal category with obvious unity $(1, I)$.

A composition law on $(A, G)$ is a morphism $(t, \gamma): G\otimes G \to G$ i.e. a transformation $\gamma: \otimes\circ (G\times G)\Rightarrow G\circ (+): |A|\times |A| \to \mathcal{C}$ i.e. a family of morphism $\gamma_{a, a'}: G(a)\otimes G(a')\to G(a+a')\ a, a'\in A$.

A graded algebra is just a monoid of $Gr.\mathcal{C}$.

Let $(A, G)$ a $A$-graded algebra considered as a strict tensor object (see $[JS]$), for define a braid on it we need a structure of 2-category on $\mathcal{C}$ (that strictly enrich the monoidal structure).

Define cells as $\rho: f\Rightarrow g: X\to Y$ as represented by a $\rho \in R$ such that $f= \rho\cdot g$, where the vertical composition correspond to the multiplication of $R$, and horizontal composition given by additive composition law.

Then a braid of the algebra $(A, G)$ is just a family of invertible elements $\epsilon(a, b)\in R,\ a, b\in A$ with $\gamma_{a, b}= \epsilon(a, b)\cdot (\gamma_{b, a}\circ tw)$ where $tw: G(a)\otimes G(b)\cong G(b)\otimes G(a)$ and such that (essentialy): $\epsilon(a, b_1+b_2)=\epsilon(a, b_1)\cdot \epsilon(a, b_2)$, $\epsilon(a_1+a_2, b)=\epsilon(a_1, b)\cdot \epsilon(a_2, b)$, and this is a symmetry if $\epsilon(a, b)^{-1}= \epsilon(b, a)$, we just find the definition of "commutation factor" of $[Bou]$ or $[GTC]$.

[Bou]: Bourbaki, Algebra chap. III (pag. 46 in the French version)

[JS] Braided Tensor Categories, Andre Joyal, Ross Street.

Advances in Mathematics
Volume 102, Issue 1, November 1993, Pages 20-78

[GTC] Graded tensor calculus, M. Scheunert. Journal of Mathematical Physics: Vol 24, No 11