定向范畴理论中的余极限
Colimits in Oriented Category Theory
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中文总结 AI 辅助
本文针对高阶范畴理论中lax余极限的几何不兼容性缺陷，提出定向余极限理论，引入兼容Gray张量积的Grothendieck构造，证明其等价性并将其应用于高阶主丛分类、伴随函子表示及Quillen定理推广。
中文摘要 AI 辅助
在高阶范畴理论中， lax余极限常被视为普通（同伦）余极限更有用、更强大的推广，普通（同伦）余极限可通过合适的局部化从lax余极限得到。然而，从几何视角看，lax余极限无法提供正确的粘合概念，它们与范畴维数、Gray张量积及其他基础几何操作不兼容。本文发展了定向余极限理论，该理论修正了lax余极限的缺陷，且在维数不超过1时与lax余极限一致。为研究定向余极限，我们引入了一种与Gray张量积上的丰富结构兼容的Grothendieck构造，证明该构造在笛卡尔纤维化与$(\fty,\fty)$-范畴预层之间诱导出一个等价关系，且该等价关系在$(\fty,\fty)$-范畴的Gray张量积上是丰富的。定向余极限同时推广了lax余极限和Gray张量积，其与lax余极限的差异，类似于Gray张量积与笛卡尔积的差异。我们通过展示高阶范畴理论中的多种基础构造并非lax余极限，而是定向余极限的实例，论证了定向余极限的必要性。作为应用，我们对高阶范畴主丛进行分类，用$(\fty,\fty)$-范畴的双笛卡尔纤维化表示高阶伴随函子，并得到了Quillen定理A和B的高阶范畴版本，这些定理在我们的框架中具有非常自然的表述。
英文摘要
In higher category theory, lax colimits are often understood to be a more useful and powerful generalization of usual (homotopy) colimits, which can be recovered from the lax colimit by a suitable localization. However, lax colimits do not provide the correct notion of gluing from the geometric perspective. Indeed, they are incompatible with the notion of categorical dimension, the Gray tensor product, and other basic geometric operations. In this paper, we develop the theory of oriented colimits, which correct the defects of lax colimits, and agree with lax colimits in dimension less than or equal to one. In order to study oriented colimits, we introduce a version of the Grothendieck construction which is compatible with enrichment in the Gray tensor product. We prove that the Grothendieck construction induces an equivalence between cartesian fibrations and presheaves of $(\infty,\infty)$-categories, which is enriched in the Gray tensor product of $(\infty,\infty)$-categories. Oriented colimits simultaneously generalize the concept of lax colimits and the Gray tensor product, and differ from lax colimits in much the same way in which the Gray tensor product differs from the cartesian product. We demonstrate the necessity of oriented colimits by showing that various fundamental constructions in higher category theory fail to be lax colimits but are instances of oriented colimits. As applications, we classify higher-categorical principal bundles, represent higher dimensional adjunctions by bicartesian fibrations of $(\infty,\infty)$-categories, and obtain higher categorical versions of Quillen's Theorems A and B, which admit very natural formulations in our framework.
