分裂双侧2-纤维的虚拟双范畴
Virtual double categories of split two-sided 2-fibrations
AI总结：
本文提出通过虚拟双范畴研究分裂双侧2-纤维，并展示其与2-函子之间的双侧Grothendieck对应。
摘要：
本文介绍了并研究了分裂双侧2-纤维和局部离散的分裂双侧2-纤维，采用形式范畴的方法。我们将Street关于分裂双侧纤维内部2-范畴的概念推广到内部sesquicategory中。给定一个sesquicategory，我们构造了一个虚拟双范畴，其水平（松散）态射是其内部的分裂双侧纤维。专门针对松散自然变换的sesquicategory，我们得到分裂双侧2-纤维的虚拟双范畴，并对其进行详细研究。然后我们限制到局部离散的分裂双侧2-纤维的子虚拟双范畴，并展示其中通常的Yoneda 2-函子满足一种双范畴形式的Yoneda态射，这种形式捕捉到与在2-范畴上Yoneda结构所满足的普遍性质相似的性质。作为结果，我们得到一个'双侧Grothendieck对应'，将局部离散的分裂双侧2-纤维A→B与2-函子B→Cat^{A^{op}}联系起来。限制到A=1，终端2-范畴，我们通过扩展其在functorial意义上的解释，改进了Buckley和Lambert对局部离散的分裂反2-纤维的'Grothendieck对应'。
英文摘要：
This paper introduces and studies split two-sided 2-fibrations and locally discrete split two-sided 2-fibrations, using a formal categorical approach. We generalise Street's notion of split two-sided fibration internal to a 2-category to one internal to a sesquicategory. Given a sesquicategory we construct a virtual double category whose horizontal (loose) morphisms are its internal split two-sided fibrations. Specialising to the sesquicategory of lax natural transformations, we obtain the virtual double category of split two-sided 2-fibrations, which we study in detail. We then restrict to the sub-virtual double category of locally discrete split two-sided 2-fibrations and show that therein the usual Yoneda 2-functors satisfy a double-categorical formal notion of Yoneda morphism, which formally captures universal properties similar to those satisfied by the morphisms comprising a Yoneda structure on a 2-category. As a consequence we obtain a 'two-sided Grothendieck correspondence' of locally discrete split two-sided 2-fibrations $A \nrightarrow B$ and 2-functors $B \to Cat^{A^{op}}$. Restricting to $A = 1$, the terminal 2-category, we improve Lambert's 'Grothendieck correspondence' for locally discrete split op-2-fibrations by extending the sense in which it is functorial.
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