逆半群上的主丛
The principal bundles over an inverse semigroup
AI总结：
本文在拓扑斯框架下研究逆半群表示，提出连通表示并证明其构成余反射子范畴，同时建立过滤函子与泛表示的等价，为逆半群表示理论提供新视角。
摘要：
本文是对拓扑斯中逆半群表示理论发展的一个贡献。它延续了Funk和Hofstra开创的工作。对于集合的拓扑斯，我们证明了逆半群S的Loganathan范畴L(S)上的无挠函子等价于S的一类特殊的非严格表示，我们称之为连通的。我们证明了后一类表示构成S的所有非严格表示范畴的一个真余反射子范畴。我们描述了L(S)上的有向且保拉回函子与S的传递且有效表示之间的对应关系，以及过滤函子与Lawson、Margolis和Steinberg引入的泛表示之间的对应关系。我们提出了在拓扑空间X上的层拓扑斯Sh(X)中逆半群的泛表示的定义，并概述了在任意拓扑斯中定义此类表示的方法。我们证明了从L(S)到拓扑斯Sh(X)的过滤函子范畴等价于S在Sh(X)中的泛表示范畴。
英文摘要：
This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra. For the topos of sets, we show that torsion-free functors on Loganathan's category $L(S)$ of an inverse semigroup $S$ are equivalent to a special class of non-strict representations of $S$, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of $S$. We describe the correspondence between directed and pullback preserving functors on $L(S)$ and transitive and effective representations of $S$, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation of an inverse semigroup in the topos of sheaves ${\mathsf{Sh}}(X)$ on a topological space $X$ as well as outline an approach on how to define such a representation in an arbitrary topos. We prove that the category of filtered functors from $L(S)$ to the topos ${\mathsf{Sh}}(X)$ is equivalent to the category of universal representations of $S$ in ${\mathsf{Sh}}(X)$.
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