Normalization of a duoidal category

Last updated Apr 23, 2024

Related.

• duoidal category
• normal duoidal category
• Normalization of profunctors over a monoidal category
• Bimodule on a bicategory

References.

• Commutativity (Garner, Lopez Franco, 2015)

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

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

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