nLab graded codifferential category




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


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


Let RR be a rig. A RR-graded monad in a category π’ž\mathcal{C} is given by a family (S r:π’žβ†’π’ž) r∈R(S_{r}:\mathcal{C} \rightarrow \mathcal{C})_{r \in R} of endofunctors, a family (m r,sA:S r(S s(A))β†’S rs(A)) r,s∈R(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:Aβ†’S 1(A)u:A \rightarrow S_{1}(A) such that:



Let RR be a rig. A RR-graded algebra modality is a RR-graded monad with a family of natural transformations βˆ‡ r,sA:S r(A)βŠ—S s(A)β†’S r+s(A)\nabla_{r,s}A:S_{r}(A) \otimes S_{s}(A) \rightarrow S_{r+s}(A) and a natural transformation Ξ· A:Iβ†’S 0(A)\eta_{A}:I \rightarrow S_{0}(A) such that for every object AA, we have that ((S r(A)) r∈R,(βˆ‡ r,s) r,s∈R,Ξ·)((S_{r}(A))_{r \in R},(\nabla_{r,s})_{r,s \in R}, \eta) is a (R,+,0)(R,+,0)-commutative graded monoid and this diagram commutes:


A RR-graded codifferential category is a CMon-enriched symmetric monoidal category with a RR-graded algebra modality and a RR-graded deriving transformation ie. a family (d rA:S r+1(A)β†’S r(A)βŠ—A)(d_{r}A:S_{r+1}(A) \rightarrow S_{r}(A) \otimes A) of natural transformation such that:

  • Linear rule:
  • Schwarz rule:

symmetric powers in a symmetric monoidal (Q plus)-linear category

differential category

Homogeneous polynomial codifferential category

Anticommutative graded codifferential category

Last revised on August 13, 2022 at 21:13:44. See the history of this page for a list of all contributions to it.