论文标题
丰富类别的oplax限制
The oplax limit of an enriched category
论文作者
论文摘要
我们表明，如果我们允许$ \ mathscr {b} $，则表格$ \ mathscr {b} \ mbox { - } \ mathbf {cat} $的2类分类，前提是我们允许$ \ mathscr {b} $可以超过bicateGories（而不是，例如，单型类别）。也就是说，对于任何$ \ Mathscr {b} $ - 类别$ \ Mathbb {X} $，我们定义A BicateGory $ \ Mathscr {B}/\ Mathbb {X} $ （\ Mathscr {B}/\ Mathbb {X}）\ Mbox { - } \ Mathbf {Cat} $。 BicateGory $ \ Mathscr {B}/\ Mathbb {X} $被描述为$ \ Mathbb {X} $的Oplax限制，被视为从混乱类别到$ \ Mathscr {b} $的Lax函数，在2级category $ \ mathbf {bicalbf {bicate} $中，bicatiC和bicate} $。从概念上，我们通过2个功能$ \ MathBf {bicat} \至2 \ Mbox { - } \ MathBf {Cat} $将每个BICATEGORY $ \ MATHSCR {B MATHSCR {B MATHSCR {B MATHSCR {B MATHSCR {B MATHSCR {b} $映射到2-Category $ \ Mathscr $ \ Mathscr {bbox {bbox} $ {cat cat cat cat cat cat cation从概念上证明了这一证明。当$ \ mathscr {b} $满足温和的局部完整性条件时，我们还表明同构$ \ Mathscr {b} \ mbox { - } \ Mathbf {cat}/\ mathbb {x} \ cong {x} \ cong （\ Mathscr {b}/\ Mathbb {X}）\ Mbox { - } \ Mathbf {Cat} $限制在$ \ Mathscr {B} \ Mbox { - } \ Mbox { - } \ Mathbf {Cat} $ \ cat $ \ m m i \ m mathbb {x} $ whos $ \ mathscr {b} \ mbox { - mbox { - mbox { - $ \ mathscr {b}/\ mathbb {x} $ - 类别承认另一个功能。
We show that 2-categories of the form $\mathscr{B}\mbox{-}\mathbf{Cat}$ are closed under slicing, provided that we allow $\mathscr{B}$ to range over bicategories (rather than, say, monoidal categories). That is, for any $\mathscr{B}$-category $\mathbb{X}$, we define a bicategory $\mathscr{B}/\mathbb{X}$ such that $\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong (\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$. The bicategory $\mathscr{B}/\mathbb{X}$ is characterized as the oplax limit of $\mathbb{X}$, regarded as a lax functor from a chaotic category to $\mathscr{B}$, in the 2-category $\mathbf{BICAT}$ of bicategories, lax functors and icons. We prove this conceptually, through limit-preservation properties of the 2-functor $\mathbf{BICAT}\to 2\mbox{-}\mathbf{CAT}$ which maps each bicategory $\mathscr{B}$ to the 2-category $\mathscr{B}\mbox{-}\mathbf{Cat}$. When $\mathscr{B}$ satisfies a mild local completeness condition, we also show that the isomorphism $\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong (\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$ restricts to a correspondence between fibrations in $\mathscr{B}\mbox{-}\mathbf{Cat}$ over $\mathbb{X}$ on the one hand, and $\mathscr{B}/\mathbb{X}$-categories admitting certain powers on the other.
