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

  • 2
    $\begingroup$ I've already such a (slightly modified but similar) definition. It's rather easy to prove that the $A$-graded objects in $\cal C$ correspond precisely to the functor category $[A,{\cal C}]$ with Day convolution monoidal product, and this might be interesting for you. Reference for this: link $\endgroup$
    – fosco
    Feb 6, 2017 at 0:23

1 Answer 1


I am not sure whether the following point of view is what you are asking for, in the sense that it is not some further development of your observations, but a seemingly different categorical description, motivated mainly by the title of the question:

The idea is that a graded (by some group), colored algebra can be equivalently described as an algebra in the braided monoidal category of representations of the group Hopf algebra, where the braiding will be uniquely determined by the color function.

To make it more clear, if $\mathbb{G}$ is a finite abelian group and $\theta:\mathbb{G}\times\mathbb{G}→C^∗$ is a skew-symmetric bicharacter (or: commutation factor or color function) on $\mathbb{G}$, then a $\mathbb{G}$-graded, $\theta$-braided Hopf algebra is defined through any of the following equivalent conditions:

  • $H$ is a $\mathbb{G}$-graded, $ϑ$-braided Hopf algebra or a $(\mathbb{G},ϑ)$-Hopf algebra.
  • $H$ is a Hopf algebra in the braided Monoidal Category $_{\mathbb{C}G}\mathcal{M}$, of representations of the group Hopf algebra $\mathbb{C}G$.
  • $H$ is a braided group for which the braiding is given by the function $ϑ:G×G→C^∗$.
  • $H$ is simultaneously an algebra, a coalgebra and a $\mathbb{C}G$-module, all its structure functions (multiplication, comultiplication, unity, counity and antipode) are $\mathbb{C}G$-module morphisms. The comultiplication $\Delta:H\rightarrow H\underline{\otimes}H$ and the counity $\epsilon:H\rightarrow C$ are algebra morphisms in the braided monoidal Category $_{\mathbb{C}G}\mathcal{M}$. ($H\underline{\otimes}H$ stands for the braided tensor product algebra). At the same time, the antipode $S:H→H$ is a “twisted” or “braided” anti-homomorphism in the sense that $S(xy)=ϑ(deg(x),deg(y))S(y)S(x)$ for any homogeneous $x,y∈H$.
  • The $\mathbb{C}G$-module $H$ is an algebra in $_{\mathbb{C}G}\mathcal{M}$ (equiv.: a $\mathbb{C}G$-module algebra) and a coalgebra in $_{\mathbb{C}G}\mathcal{M}$ (equiv.: a $\mathbb{C}G$-module coalgebra), the comultiplication and the counity are algebra morphisms in the braided monoidal Category $_{\mathbb{C}G}\mathcal{M}$ and at the same time, the antipode is an algebra anti-homomorphism in the braided monoidal Category $_{\mathbb{C}G}\mathcal{M}$.

As an example, consider the Universal Enveloping algebras (UEA) of the $ϑ$-colored $G$-graded Lie algebras (or $(G,ϑ)$-Lie algebras or Lie algebras in the braided monoidal category $_{\mathbb{C}G}\mathcal{M}$): these are $G$-graded Hopf algebras or to be more rigorous $(G,ϑ)$-Hopf algebras or $G$-graded, $ϑ$-braided Hopf algebras. In this last case, $ϑ:G×G→C^∗$ stands for a skew-symmetric bicharacter on $G$ (or color function or commutation factor according to different authors), which is equivalent to a triangular universal $R$ -matrix on the group Hopf algebra $\mathbb{C}G$. This finally entails a symmetric braiding in the Monoidal Category $_{\mathbb{C}G}\mathcal{M}$ of the modules over the group Hopf algebra $\mathbb{C}G$.
(In fact, there is a bijection, from the set of bicharacters of a finite abelian group $\mathbb{G}$ onto the set of Universal $R$-matrices of the group Hopf algebra $\mathbb{C}G$ and from there onto the set of the braidings of the monoidal Category of representations $_{\mathbb{C}G}\mathcal{M}$:
The above described bijection, is such that the skew-symmetric bicharacters (i.e., the color functions or commutation factors) are mapped onto triangular universal $R$ -matrices and thus onto symmetric braidings of the symmetric monoidal category $_{\mathbb{C}G}\mathcal{M}$.)

The preceding description easily generalizes for a non-abelian group: one has just to replace the braided monoidal category $_{\mathbb{C}G}\mathcal{M}$ of the $\mathbb{C}G$-modules with the braided monoidal category $\mathcal{M}^{\mathbb{C}G}$ of the comodules of the group hopf algebra $\mathbb{C}G$. In this case, a bicharacter on $G$ is essentially the same thing with a coquasitriangular structure on $\mathbb{C}G$. (In the abelian case, the situation simplifies using the isomorphism of the coresponding categories of modules and comodules: $_{\mathbb{C}G}\mathcal{M}\cong\mathcal{M}^{\mathbb{C}G}$).

If you are interested in this point of view, you can have a look at the section 3.1 of this article and the references therein. (see particularly: arXiv:q-alg/9508016)


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