topos的局部Istropy群
The localic Istropy group of a topos
摘要：
J.Funk，P.Hofstra和B.Steinberg证明了任何Grothendieck topos T都被赋予一个标准群对象，称为它的迷向群，它函符地作用在T的每个对象上。我们证明了这个群实际上是局部群对象的点群，称为局部各向同性群，它也作用于每个对象，实际上也作用于每个内部局部和每个T-topos。这个新的局域各向同性群比原来的版本具有更好的功能性和稳定性，并揭示了一些现象，观察到的普通各向同性群的高各向同性。我们证明，特别是使用一个本地版本的各向同性商，任何几何态射都可以分解为唯一的连接原子几何态射其次是一个所谓的“本质上各向异性”的几何态射，连接原子态射正是由一个开放的各向同性行动商。
英文摘要：
It has been shown by J.Funk, P.Hofstra and B.Steinberg that any Grothendieck topos T is endowed with a canonical group object, called its isotropy group, which acts functorially on every object of T. We show that this group is in fact the group of points of a localic group object, called the localic isotropy group, which also acts on every object, and in fact also on every internal locales and on every T-topos. This new localic isotropy group has better functoriality and stability property than the original version and shed some lights on the phenomenon of higher isotropy observed for the ordinary isotropy group. We prove in particular using a localic version of the isotropy quotient that any geometric morphism can be factored uniquely as a connected atomic geometric morphism followed by a so called "essentially anisotropic" geometric morphism, and that connected atomic morphism are exactly the quotient by an open isotropy action.
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