Normalization of a duoidal category

Related.

• duoidal category
• normal duoidal category
• Normalization of profunctors over a monoidal category
• Bimodule on a bicategory
• Par is the duoidal normalization of Set

References.

• Commutativity (Garner, Lopez Franco, 2015)

Extra notes

Let $M$ be a bimonoid in the duoidal category $(\mathbb{V},\otimes ,I,◁,N)$, with maps $e﹕I\to M$ and $m﹕M\otimes M\to M$; and with maps $u﹕M\to N$ and $d﹕M\to M◁M$. Consider the category of two-sided $M^{\otimes }$-modules; this category has a monoidal structure lifted from $(\mathbb{V},◁,N)$:

• the unit, $N$, has a module structure with $M\otimes N\otimes M\stackrel{u\otimes id\otimes u}{⟶}N\otimes N\otimes N⟶N;$
• the sequencing of two $M^{\otimes }$-modules is a $M^{\otimes }$-module with $M\otimes (A◁B)\otimes M\to (M◁M)\otimes (A◁B)\otimes (M◁M)\to (M\otimes A\otimes M)◁(M\otimes B\otimes M)\to A◁B.$ Moreover, whenever $\mathbb{V}$ admits reflexive coequalisers preserved by $(\otimes )$, the category of $M^{\otimes }$-bimodules is monoidal with the tensor of bimodules: the coequaliser $A\otimes M\otimes B\rightrightarrows A\otimes B\twoheadrightarrow A{\otimes }_{M}B.$ In this case $({\mathbf{Bimod}}_M^{\otimes },{\otimes }_M,M,◁,N)$ is a duoidal category.

Definition. Let $(\mathbb{V},\otimes ,I,◁,N)$ be a duoidal category with reflexive coequalisers preserved by $(\otimes )$. The normalization (Garner, Lopez Franco, 2015) of $\mathbb{V}$ is the normal duoidal category $\mathcal{N}(\mathbb{V})=({\mathbf{Bimod}}_N^{\otimes },{\otimes }_N,N,◁,N).$