nLab algebra modality

Idea

The notion of algebra modality is used to define codifferential categories. However, it could be use in other types of categorical doctrines.

Definition

Definition

An algebra modality in a symmetric monoidal category $(\mathcal{C}, \otimes, I)$ is given by a monad $(S,m,u)$ and two natural transformations $\eta:I \rightarrow S(A)$ and $\nabla:S(A) \otimes S(A) \rightarrow S(A)$ such that for every $A \in \mathcal{C}$, $(S(A), \nabla, \eta)$ is a commutative monoid in $(\mathcal{C},\otimes,I)$ and this diagram commutes:

Properties

Proposition

If $R$ is a commutative rig, then the symmetric algebra defines an algebra modality in $Mod_{R}$.

• $Sym(A)$ is the symmetric algebra of the module $A$.
• We have $\nabla_{A}:Sym(A) \otimes Sym(A) \rightarrow Sym(A)$.
• We have $\eta_{A}:A \rightarrow Sym(A)$.
• The unit $A \rightarrow Sym(A)$ of the monad is just the injection $x \mapsto x$.
• The multiplication $Sym(Sym(A)) \rightarrow Sym(A)$ of the monad is given on pure tensors by $(x_{1}^{(1)} \otimes_{s} ... \otimes_{s} x_{n_{1}}^{(1)}) \boxtimes_{s} ... \boxtimes_{s} (x_{1}^{(p)} \otimes_{s} ... \otimes_{s} x_{n_{p}}^{(p)}) \mapsto x_{1}^{(1)} \otimes_{s} ... \otimes_{s} x_{n_{1}}^{(1)} \otimes ... \otimes x_{1}^{(p)} \otimes_{s} ... \otimes_{s} x_{n_{p}}^{(p)}$. It is a kind of composition of polynomials.

Proposition

Commutative rigs are exactly the algebras (over a monad) over the algebra modality $Sym$ in the category of commutative monoids.

References

Algebra modalities are part of the definition of a codifferential category ie. a category $\mathcal{C}$ such that $\mathcal{C}^{op}$ is a differential category.

Differential categories were introduced in:

• R. F. Blute, J. R. B. Cockett, and R. A. G. Seely: Differential categories. Math. Struct. Comput. Sci. 16(06), 1049–1083 (2006) (doi:10.1017/S0960129506005676)

and revised in:

• Richard Blute, Robin Cockett, Jean-Simon Lemay and Robert Seely, Differential categories revisited, Appl. Categ. Struct. 28, 171-235 (2020). (arXiv:1806.04804, doi:10.1007/s10485-019-09572-y)