A classifying localic category for locally compact locales with application to the Axiom of Infinity (poster)

Christopher Francis Townsend
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Abstract

For an internal category $\mathbb{C}$ in a cartesian category $\mathcal{C}$ we define, naturally in objects $X$ of $\mathcal{C}$, $Prin_{\mathbb{C}}(X)$. This is a category whose objects are principal $c \mathbb{C}$-bundles over $X$ and whose morphisms are principal $c(\mathbb{C}^{\uparrow})$-bundles. Here $c(\_)$ denotes taking the core groupoid of a category (same objects but only isomorphisms as morphisms) and $\mathbb{C}^{\uparrow}$ is the arrow category of $\mathbb{C}$ (objects morphisms, morphisms commuting squares). We show that $X \mapsto Prin_{\mathbb{C}}(X)$ is a stack of categories and call stacks of this sort lax-geometric. We then provide two sufficient conditions for a stack to be lax-geometric and use them to prove that the pseudo-functor $X \mapsto \mathbf{LK}_{Sh(X)}$ on the category of locales $\mathbf{Loc}$ is a lax-geometric stack. Here $\mathbf{LK}_{Sh(X)}$ is the category of locally compact locales in the topos of sheaves over $X$, $Sh(X)$. Therefore there exists a localic category $\mathbb{C}_{\mathbf{LK}}$ such that $\mathbf{LK}_{Sh(X)} \simeq Prin_{\mathbb{C}_{\mathbf{LK}}}(X)$ naturally for every locale $X$. We then show how this can be used to give a new localic characterisation of the Axiom of Infinity.
局部紧凑局部的分类局部范畴及其在无穷公理中的应用(海报)
对于笛卡尔范畴$\mathcal{C}$中的内部范畴$\mathbb{C}$,我们在$\mathcal{C}$的对象$X$中自然地定义了$Prin_{\mathbb{C}}(X)$.这是一个其对象是在$X$上的主$c \mathbb{C}$束,其形态是主$c(\mathbb{C}^{\uparrow})$束的范畴。这里$c(\_)$表示取一个范畴的核心群(对象相同,但只有同态作为态),而$\mathbb{C}^{\uparrow}$是$\mathbb{C}$的箭范畴(对象态,态相交平方)。我们证明 $X\mapsto Prin_{\mathbb{C}}(X)$ 是一个范畴堆栈,并称这种堆栈为宽松几何堆栈。然后,我们提供了一个栈为ax-几何栈的两个充分条件,并用它们证明了本地范畴 $\mathbf{Loc}$ 上的伪矢量 $X \mapsto\mathbf{LK}_{Sh(X)}$ 是ax- 几何栈。这里的$mathbf{LK}_{Sh(X)}$是$X$上的剪切拓扑中的局部紧密局部范畴,即$Sh(X)$。因此,存在一个局部范畴 $\mathbb{C}_{\mathbf{LK}}$ ,使得$\mathbf{LK}_{Sh(X)} $ \simeq Prin_{\mathbb{C}_{\mathbf{LK}}}(X)$ 自然永存局部 $X$。然后我们将展示如何利用这一点给出无穷公理的新局部特性.
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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