Mario Román


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Normalization of profunctors over a monoidal category

Last updated Sep 8, 2023

Let $(ℂ,⊗,I)$ be a monoidal category. The category of endoprofunctors $ℂ^{op} × ℂ → \mathbf{Set}$ is duoidal with composition $(◁)$ and Day convolution $(⊛)$. $$(ℂ^{op}×ℂ,\mathbf{Set}, ⊛, I, ◁, \hom).$$ The category of $\hom^⊛$-bimodules has been traditionally called the category of Tambara modules. Their tensoring is possible because $[ℂ{op}×ℂ,\mathbf{Set}]$ admits reflexive coequalisers. $$\mathcal{N}(ℂ^{op}×ℂ,\mathbf{Set}, ⊛, I, ◁, \hom) = (\mathbf{Tamb}, ⊛_{\hom}, \hom, ◁, \hom).$$Theorem. The category of Tambara modules is a normal duoidal category and, in fact, it is the normalization of the duoidal category of endoprofunctors. normalization-of-profunctors-over-a-monoidal-category

Normalization is the process that transforms wires into boards, see Tambara modules as algebras.

Tags: Tambara module, duoidal normalization.

References: I believe this result, in this formulation, is novel. For reference on the normalization of a duoidal category, see Commutativity (Garner, Lopez Franco, 2015).