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

2-Filteredness and The Point of Every Galois Topos

A connected locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bi-limits of topoi, we show that every Galois topos has a point.Fil: Dubuc, Eduardo Julio. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin