Freely adjoining monoidal duals

Abstract

Given a monoidal category $\mathcal{C}$ with an object $J$, we construct a monoidal category $\mathcal{C}[J^{\vee }]$ by freely adjoining a right dual $J^{\vee }$ to $J$. We show that the canonical strong monoidal functor $\Omega :\mathcal{C}\to \mathcal{C}[J^{\vee }]$ provides the unit for a biadjunction with the forgetful 2-functor from the 2-category of monoidal categories with a distinguished dual pair to the 2-category of monoidal categories with a distinguished object. We show that $\Omega :\mathcal{C}\to \mathcal{C}[J^{\vee }]$ is fully faithful and provide coend formulas for homs of the form $\mathcal{C}[J^{\vee }](U,\Omega A)$ and $\mathcal{C}[J^{\vee }](\Omega A,U)$ for $A\in \mathcal{C}$ and $U\in \mathcal{C}[J^{\vee }]$. If $\mathbb{N}$ denotes the free strict monoidal category on a single generating object $1$ then $\mathbb{N}[1^{\vee }]$ is the free monoidal category $\mathrm{Dpr}$ containing a dual pair $-⊣+$ of objects. As we have the monoidal pseudopushout $\mathcal{C}[J^{\vee }]\simeq \mathrm{Dpr}{+}_{\mathbb{N}}\mathcal{C}$, it is of interest to have an explicit model of $\mathrm{Dpr}$: we provide both geometric and combinatorial models. We show that the (algebraist's) simplicial category $\Delta$ is a monoidal full subcategory of $\mathrm{Dpr}$ and explain the relationship with the free 2-category $\mathrm{Adj}$ containing an adjunction. We describe a generalization of $\mathrm{Dpr}$ which includes, for example, a combinatorial model $\mathrm{Dseq}$ for the free monoidal category containing a duality sequence $X_0⊣X_1⊣X_2⊣\dots$ of objects. Actually, $\mathrm{Dpr}$ is a monoidal full subcategory of $\mathrm{Dseq}$.

27 pages

References in corpus (1)

• Autonomization of Monoidal Categories