论文标题
单体中心和类固定类别类别
Monoidal centres and groupoid-graded categories
论文作者
论文摘要
我们用$ \ mathrm {mod} $表示类别之间的双面模块（也称为分配器，双模型和分销商）的单层生物;张量产品是类别的笛卡尔产品。对于一个groupoid $ \ scr {g} $，我们研究单型中心$ \ mathrm {zps}（\ scr {g}，\ mathrm {modrm {mod}^{\ mathrm {op {op}}）$ $ \ mathrm {ps}（\ scr {g}，\ mathrm {mod}^{\ mathrm {op}}）$ of pseudofunctors和pseudonatural transformations;张量产品是点的。 Alexei Davydov在单体类别中定义了一个单体的完整中心。我们定义了一个更高维的版本：在单型bicategory $ \ scr {m} $中的单形壁（= pseudomonoid）的完整单体中心，它是单型中心$ \ mathrm {z} \ scr {m} $ $ \ s $ \ scr {m scr {m} $的单型中心$ \ mathrm {z} \ scr {m} $。每个纤维$π：\ scr {h} \ to groupoids之间的\ to \ scr {g} $提供了一个$ \ mathrm {ps}（\ scr {g}，\ scr {g}，\ mathrm {mod}^{\ mathrm {\ mathrm {\ mathrm {op {op}}}的完整单体中心的示例。对于一组$ g $，我们解释了Turaev和Virelizier考虑的$ G $分类结构，以构建拓扑不变性，适合这种单型生物的环境。我们看到它们的结构是$ g $ ACTS的$ K $ - 线性类别的单型生物中心的单叶层。
We denote the monoidal bicategory of two-sided modules (also called profunctors, bimodules and distributors) between categories by $\mathrm{Mod}$; the tensor product is cartesian product of categories. For a groupoid $\scr{G}$, we study the monoidal centre $\mathrm{ZPs}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of the monoidal bicategory $\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of pseudofunctors and pseudonatural transformations; the tensor product is pointwise. Alexei Davydov defined the full centre of a monoid in a monoidal category. We define a higher dimensional version: the full monoidal centre of a monoidale (= pseudomonoid) in a monoidal bicategory $\scr{M}$, and it is a braided monoidale in the monoidal centre $\mathrm{Z}\scr{M}$ of $\scr{M}$. Each fibration $π: \scr{H} \to \scr{G}$ between groupoids provides an example of a full monoidal centre of a monoidale in $\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$. For a group $G$, we explain how the $G$-graded categorical structures, as considered by Turaev and Virelizier in order to construct topological invariants, fit into this monoidal bicategory context. We see that their structures are monoidales in the monoidal centre of the monoidal bicategory of $k$-linear categories on which $G$ acts.
