2-filteredness and the point of every Galois topos

论连通 Grothendieck 拓扑斯的 profinite 基本群

On the profinite fundamental group of a connected Grothendieck topos

本文通过连通 Grothendieck 拓扑斯中有限对象张成的 Boolean pretopos 及其本质唯一 Galois 点的自同构群，给出了拓扑斯基本群的内在定义，并证明了有限生成情形下的分类等价性。

Admissibility, stable units and connected components

Consider a reflection from a finitely-complete category $\mathbb{C}$ into its full subcategory $\mathbb{M}$, with unit $η:1_\mathbb{C}\rightarrow HI$. Suppose there is a left-exact functor $U$ into the category of sets, such that $UH$ reflects isomorphisms and $U(η_C)$ is a surjection, for every $C\in\mathbb{C}$. If, in addition, all the maps $\mathbb{M}(T,M)\rightarrow \mathbf{Set}(1,U(M))$ induced by the functor $UH$ are surjections, where $T$ and 1 are respectively terminal objects in $\mathbb{C}$ and $\mathbf{Set}$, for every object $M$ in the full subcategory $\mathbb{M}$, then it is true…

The fundamental progroupoid of a general topos

It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the \emph{fundamental progroupoid}, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be…