A graded codifferential category is like a codifferential category but where the algebra modality is replaced by a graded algebra modality. Whereas $FVec_{\mathbb{K}}$ is only a trivial codifferential category, with all the structure equal to zero, $FVec_{\mathbb{K}}$ is a non-trivial codifferential category.

Definition

We will use graded monads with grading in the multiplicative monoid of a commutative rig. Also, the term rig must be read as βcommutative rigβ below.

Definition

Let $R$ be a rig. A $R$-graded monad in a category $\mathcal{C}$ is given by a family $(S_{r}:\mathcal{C} \rightarrow \mathcal{C})_{r \in R}$ of endofunctors, a family $(m_{r,s}A:S_{r}(S_{s}(A)) \rightarrow S_{rs}(A))_{r,s \in R}$ of natural transformations and a natural transformation $u:A \rightarrow S_{1}(A)$ such that:

and

Definition

Let $R$ be a rig. A $R$-graded algebra modality is a $R$-graded monad with a family of natural transformations $\nabla_{r,s}A:S_{r}(A) \otimes S_{s}(A) \rightarrow S_{r+s}(A)$ and a natural transformation $\eta_{A}:I \rightarrow S_{0}(A)$ such that for every object $A$, we have that $((S_{r}(A))_{r \in R},(\nabla_{r,s})_{r,s \in R}, \eta)$ is a $(R,+,0)$-commutative graded monoid and this diagram commutes:

Definition

A $R$-graded codifferential category is a CMon-enriched symmetric monoidal category with a $R$-graded algebra modality and a $R$-graded deriving transformation ie. a family $(d_{r}A:S_{r+1}(A) \rightarrow S_{r}(A) \otimes A)$ of natural transformation such that: