松弛逗号$2$-范畴与容许$2$-函子
Lax comma $2$-categories and admissible $2$-functors
AI总结：
本文开展Janelidze-Galois理论的二维扩展研究，提出松弛幂等语境下的2-容许2-函子概念，研究松弛逗号2-范畴性质，证明相关2-伴随诱导关系并给出充分条件与实例，最后关联余等化子与Kan扩张。
摘要：
本文是对Janelidze-Galois理论基本思想与结果的二维扩展研究的一项贡献。文中针对松弛幂等语境，提出了与“绝对容许Galois结构”相对应的合适概念，该概念与“松弛正交分解系统”的语境相兼容。作为研究工作的一部分，我们探究了松弛逗号2-范畴，给出了与通常逗号范畴基本性质相对应的类比结果。我们证明，一个2-范畴的每个态射都会在松弛逗号2-范畴与逗号2-范畴之间诱导出一个2-伴随，其作用相当于通常的“基变换函子”。借助这些诱导出的2-伴随，我们得以证明每个2-伴随都会在松弛逗号2-范畴与逗号2-范畴之间诱导出多个2-伴随，这便是我们所提出的、对应于Janelidze-Galois理论中常用的逗号范畴提升的类比概念。我们给出了使这些提升具有2-前单子性并诱导出松弛幂等2-单子的充分条件，该松弛幂等2-单子正对应我们提出的2-容许2-函子概念。为开展这项研究，我们分析了2-伴随的合成何时为松弛幂等2-单子，以及何时具有2-前单子性。随后我们给出了2-容许2-函子（尤其是简单2-函子）的实例，特别用到了一项结论：所有经典意义下的容许（2-）函子也都是2-容许的（因而也是简单的）。本文最后建立了松弛逗号2-范畴中的余等化子与Kan扩张之间的关联。
英文摘要：
This paper is a contribution towards a two dimensional extension of the basic ideas and results of Janelidze-Galois theory. In the present paper, we give a suitable counterpart notion to that of \textit{absolute admissible Galois structure} for the lax idempotent context, compatible with the context of \textit{lax orthogonal factorization systems}. As part of this work, we study lax comma $2$-categories, giving analogue results to the basic properties of the usual comma categories. We show that each morphism of a $2$-category induces a $2$-adjunction between lax comma $2$-categories and comma $2$-categories, playing the role of the usual \textit{change of base functors}. With these induced $2$-adjunctions, we are able to show that each $2$-adjunction induces $2$-adjunctions between lax comma $2$-categories and comma $2$-categories, which are our analogues of the usual lifting to the comma categories used in Janelidze-Galois theory. We give sufficient conditions under which these liftings are $2$-premonadic and induce a lax idempotent $2$-monad, which corresponds to our notion of $2$-admissible $2$-functor. In order to carry out this work, we analyse when a composition of $2$-adjunctions is a lax idempotent $2$-monad, and when it is $2$-premonadic. We give then examples of our $2$-admissible $2$-functors (and, in particular, simple $2$-functors), specially using a result that says that all admissible ($2$-)functors in the classical sense are also $2$-admissible (and hence simple as well). We finish the paper relating coequalizers in lax comma $2$-categories and Kan extensions.
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