论文标题
自由毗邻的单双二元
Freely adjoining monoidal duals
论文作者
论文摘要
给定一个带有对象$ j $的单型类别$ \ mathscr {c} $，我们通过自由毗邻右二$ j^{\ vee} $ to $ j $来构建一个单型类别$ \ mathscr {c} [j^{\ vee}] $。我们表明，规范强的单体函数$ω：\ Mathscr {C} \ to \ Mathscr {C} [C} [j^{\ VEE}] $为单型型单体类别与杰出的偶数偶数类别的单型偶数类别提供了与2类互动的偶数对偶数对偶数的单位相分为2级的单型偶数的单位，以2级差异为2类别，以2类别为单位。我们表明$ω：\ mathscr {c} \ to \ mathscr {c} [j^{\ vee}] $是完全忠实的，并为形式的homs提供了coend cormulas $ \ mathscr {c} [c} [j^{j^{\ vee}]（u，ωa）$和$，ωa）$ $ \ mathscr {c} [j^{\ vee}]（ωa，u）$ for $ a \ in \ mathscr {c} $和$ u \ in \ mathscr {c} [j^{\ vee}] $。 如果$ \ mathbb {n} $表示单个生成对象上的免费严格单体类别$ 1 $，则$ \ mathbb {n} [1^{\ vee}] $是免费的单体类别$ \ mathrm {dpr} $，其中包含Dual Pair Pair Pair Pair $ - \ dashv +$ $ $ of Objects。因为我们具有单型假ushout $ \ mathscr {c} [j^{\ vee}] \ simeq \ simeq \ mathrm {dpr} +_ {\ mathbb {n}}}} \ mathscr {c} $，这是$ \ nater $ \ math的典型模型}型号。我们表明（代数）的简单类别$δ$是$ \ mathrm {dpr} $的单型完整子类别，并解释了与免费的2类别$ \ mathrm {adj aff} $包含相邻的关系。我们描述了$ \ mathrm {dpr} $的概括，其中包括一个组合模型$ \ mathrm {dseq} $，用于免费的单型类别，其中包含偶性序列$ x_0 \ dashv x_1 \ dashv x_1 \ dashv x_2 x_2 x_2 \ dashv \ dashv \ dashv \ dashv \ dots $。实际上，$ \ mathrm {dpr} $是$ \ mathrm {dseq} $的单型完整子类别。
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 $Ω: \mathscr{C}\to \mathscr{C}[J^{\vee}]$ 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 $Ω: \mathscr{C}\to \mathscr{C}[J^{\vee}]$ is fully faithful and provide coend formulas for homs of the form $\mathscr{C}[J^{\vee}](U,ΩA)$ and $\mathscr{C}[J^{\vee}](ΩA,U)$ 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 $Δ$ 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 $\mathrm{Dseq}$.
