论文标题
Lax Comma $ 2 $ - 类别和可允许的$ 2 $ - 调解器
Lax comma $2$-categories and admissible $2$-functors
论文作者
论文摘要
本文是对Janelidze-Galois理论的基本思想和结果的二维扩展的贡献。在本文中，我们给出了与\ textit {绝对可允许的galois结构}的合适对应概念，该概念与\ textit {lax orthoconal ofthoconal分解系统}的上下文兼容。作为这项工作的一部分，我们研究Lax Comma $ 2 $类别，从而为通常的逗号类别的基本属性提供了模拟结果。我们表明，$ 2 $类别的每种形态都会引起$ 2 $ 2 $ 2 $ 2 $ - 类别和逗号$ 2 $ - 类别之间的$ 2 $ - 类别，扮演通常的\ textit {Change of base fuctors}的角色。有了这些引起的$ 2 $ - 调整，我们能够证明，每$ 2 $ - adjuntion诱发了$ 2 $ 2 $ - 逗号$ 2 $ 2 $ - 类别和逗号$ 2 $ - 类别，这是我们对Janelidze-galois neverys in Janelidze-galois eyse中使用的相似之处。我们提供了足够的条件，这些升降机为$ 2 $ - 普雷米纳尼（Premonadic），并引起了宽松的$ 2 $ -Monad，这对应于我们的概念$ 2 $ - 可加入的$ 2 $ - 功能。为了进行这项工作，我们分析了$ 2 $ adjunctions的成分是一种宽松的$ 2 $ -MONAD，并且何时为$ 2 $ -PREMONADIC。然后，我们提供了我们$ 2 $可加入的$ 2 $ functors（尤其是简单的$ 2 $ functors）的示例，特别是使用结果，表明所有可接受的（$ 2 $ - ）在经典意义上也是$ 2 $ -2 $ -Admissible（也是简单）。我们完成了与Lax Comma $ 2 $类别和KAN扩展相关的纸张相关的纸张。
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.
