Freely adjoining monoidal duals

Abstract

Given a monoidal category $\mathscr{C}$ with an object $J$, we construct a monoidal category $\mathscr{C}[J^{\vee}]$ by freely adjoining a right dual $J^{\vee}$ to $J$. We show that the canonical strong monoidal functor $\Omega : \mathscr{C}\to \mathscr{C}[J^{\vee}]$ 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 $\Omega : \mathscr{C}\to \mathscr{C}[J^{\vee}]$ is fully faithful and provide coend formulas for homs of the form $\mathscr{C}[J^{\vee}](U,\Omega A)$ and $\mathscr{C}[J^{\vee}](\Omega A,U)$ for $A\in \mathscr{C}$ and $U\in \mathscr{C}[J^{\vee}]$. 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 $- \dashv +$ of objects. As we have the monoidal pseudopushout $\mathscr{C}[J^{\vee}] \simeq \mathrm{Dpr} +_{\mathbb{N}} \mathscr{C}$, 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 $X_0\dashv X_1\dashv X_2 \dashv \dots$ of objects. Actually, $\mathrm{Dpr}$ is a monoidal full subcategory of $\mathrm{Dseq}$.