879 research outputs found
On the representation theory of Galois and Atomic Topoi
We elaborate on the representation theorems of topoi as topoi of discrete
actions of various kinds of localic groups and groupoids. We introduce the
concept of "proessential point" and use it to give a new characterization of
pointed Galois topoi. We establish a hierarchy of connected topoi:
[1. essentially pointed Atomic = locally simply connected],
[2. proessentially pointed Atomic = pointed Galois],
[3. pointed Atomic].
These topoi are the classifying topos of, respectively: 1. discrete groups,
2. prodiscrete localic groups, and 3. general localic groups.
We analyze also the unpoited version, and show that for a Galois topos, may
be pointless, the corresponding groupoid can also be considered, in a sense,
the groupoid of "points". In the unpointed theories, these topoi classify,
respectively: 1. connected discrete groupoids, 2. connected (may be pointless)
prodiscrete localic groupoids, and 3. connected groupoids with discrete space
of objects and general localic spaces of hom-sets, when the topos has points
(we do not know the class of localic groupoids that correspond to pointless
connected atomic topoi).
We comment and develop on Grothendieck's galois theory and its generalization
by Joyal-Tierney, and work by other authors on these theories.Comment: This is a revised version of arXiv.org/math.CT/02008222 to appear in
JPA
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
considered as covering projections. The key contribution of this paper is a
novel definition of covering projection for a general topos, which coincides
with the usual definition when the topos is locally connected. The results in
this paper were presented in a talk at the Category Theory Conference,
Vancouver July 2004.Comment: 19 page
A construction of 2-cofiltered bilimits of topoi
We show the existence of bilimits of 2-cofiltered diagrams of topoi,
generalizing the construction of cofiltered bilimits developed in "SGA 4
Springer LNM 270 (1972)". For any given such diagram, we show that it can be
represented by a 2-cofiltered diagram of small sites with finite limits, and we
construct a small site for the inverse limit topos. This is done by taking the
2-filtered bicolimit of the underlying categories and inverse image functors.
We use the construction of this bicolimit developed in "A construction of
2-filtered bicolimits of categories, Cah. Top. et Geo. Diff. Vol. XLVII-2
(2006)", where it is proved that if the categories in the diagram have finite
limits and the transition functors are exact, then the bicolimit category has
finite limits and the pseudocone functors are exact. An application of our
result here is the fact that every Galois topos has points "2-Filteredness and
the point of every Galois topos, Proc. CT2007, App. Cat. St., Vol. 18, 2,
(2010)".Comment: 7 page
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