论文标题
增强虚拟双重类别的形式类别理论
Formal category theory in augmented virtual double categories
论文作者
论文摘要
在本文中，我们在增强的虚拟双重类别中开发了形式类别理论。值得注意的是，我们正式化了kan扩展的经典概念，yoneda嵌入$ \ text y_a \ colon a \ colon a \ to \ hat a $，精确方，总类别，总类别和'Small'Cococoction;后者在适当的意义上。在整个过程中，我们将形式化与它们相应的$ 2 $分类对应物进行了比较。我们的方法有几个优势。例如，增强虚拟双重类别的结构自然使我们能够隔离条件，以确保正式的预局部对象的小相结合$ \ hat a $。 Given a monoidal augmented virtual double category $\mathcal K$ with a Yoneda embedding $\text y_I \colon I \to \hat I$ for its monoidal unit $I$ we prove that, for any 'unital' object $A$ in $\mathcal K$ that has a 'horizontal dual' $A^\circ$, the Yoneda embedding $\text y_A \ colon a \ to \ hat a $在且仅当存在“ innof” $ [a^\ circ，\ hat i] $时。 This result is a special case of a more general result that, given a functor $F\colon \mathcal K \to \mathcal L$ of augmented virtual double categories, allows a Yoneda embedding in $\mathcal L$ to be "lifted", along a pair of 'universal morphisms' in $\mathcal L$, to a Yoneda embedding in $\mathcal K$.
In this article we develop formal category theory within augmented virtual double categories. Notably we formalise the classical notions of Kan extension, Yoneda embedding $\text y_A\colon A \to \hat A$, exact square, total category and 'small' cocompletion; the latter in an appropriate sense. Throughout we compare our formalisations to their corresponding $2$-categorical counterparts. Our approach has several advantages. For instance, the structure of augmented virtual double categories naturally allows us to isolate conditions that ensure small cocompleteness of formal presheaf objects $\hat A$. Given a monoidal augmented virtual double category $\mathcal K$ with a Yoneda embedding $\text y_I \colon I \to \hat I$ for its monoidal unit $I$ we prove that, for any 'unital' object $A$ in $\mathcal K$ that has a 'horizontal dual' $A^\circ$, the Yoneda embedding $\text y_A \colon A \to \hat A$ exists if and only if the 'inner hom' $[A^\circ, \hat I]$ exists. This result is a special case of a more general result that, given a functor $F\colon \mathcal K \to \mathcal L$ of augmented virtual double categories, allows a Yoneda embedding in $\mathcal L$ to be "lifted", along a pair of 'universal morphisms' in $\mathcal L$, to a Yoneda embedding in $\mathcal K$.
