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 ${\rm{N}}$ denotes the free strict monoidal category on a single generating object 1, then ${\rm{N[}}{{\rm{1}}^ \vee }{\rm{]}}$ is the free monoidal category Dpr containing a dual pair – ˧ + of objects. As we have the monoidal pseudopushout $\mathcal C[{J^ \vee }] \simeq {\rm{Dpr}}{{\rm{ + }}_{\rm{N}}}\mathcal C$, it is of interest to have an explicit model of Dpr: we provide both geometric and combinatorial models. We show that the (algebraist’s) simplicial category Δ is a monoidal full subcategory of Dpr and explain the relationship with the free 2-category Adj containing an adjunction. We describe a generalization of Dpr which includes, for example, a combinatorial model Dseq for the free monoidal category containing a duality sequence X0 ˧ X1 ˧ X2 ˧ … of objects. Actually, Dpr is a monoidal full subcategory of Dseq.

给定一个带有对象J的幺半群类别$\mathcal C$ ，我们通过自由地将一个右对偶${J^ \vee }$连接到J来构造一个幺半群类别$\mathcal C[{J^ \vee }]$。我们证明了典型的强幺半群函子$\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$提供了与来自 2 类幺半群类别的健忘 2 函子的双向联结的单位具有显着对象的 2 类幺半群类别的显着对偶。我们证明$\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$是完全忠实的，并为$\mathcal C[{J^ \vee }](U ,\,\欧米茄 A)$和$\mathcal C[{J^ \vee }](\Omega A,U)$对于$A \in \mathcal C$和$U \in \mathcal C[{J^ \vee }]$。如果${\rm{N}}$表示单个生成对象 1 上的自由严格幺半群类别，则${\rm{N[}}{{\rm{1}}^ \vee }{\rm{] }}$是自由幺半群类别 Dpr，包含对偶对 - ˧ + 对象。因为我们有幺半群赝推$\mathcal C[{J^ \vee }] \simeq {\rm{Dpr}}{{\rm{ + }}_{\rm{N}}}\mathcal C$，有一个 Dpr 的显式模型很有趣：我们提供几何模型和组合模型。我们证明（代数家的）单纯类别 Δ 是 Dpr 的一个单面全子类别，并解释了与包含一个附加词的自由 2 类别 Adj 的关系。我们描述了 Dpr 的泛化，其中包括，例如，包含对象的对偶序列X 0 ˧ X 1 ˧ X 2 ˧ ... 的自由幺半群类别的组合模型 Dseq。实际上，Dpr 是 Dseq 的一个幺半群全子类。