For any small category, C, the presheaf category of functors from C^op to Set is a topos (and given a coverage on C restricting to those functors that are sheaves gives rise to a topos). Are there any known results of this type where C is not necessarily small?
A necessary condition for Psh(C) being a topos is that for every object I of C the slice category C/I is small because this ensures that Sub(C/I) is small and thus Omega exists. A typical such example is the big poset Ord of all ordinals. But it is perhaps more interesting to consider the following ``partial topos'' over Ord consisting of all presheaves A over Ord whose support supp(A) := {I \in C | A(I)} is inhabited. This category has the following distinguishing properties: (1) every slice is a (presheaf) topos (2) the full subcategory of subterminals is reflective and filtered. Categories having these properties have been introduced and studied by Jean Benabou under the name PARTIAL TOPOS. A ``pocket version'' of the above is the category of finite trees consisting of those presheaves over \omega which have finite support and take values in FinSet, the category of finite set. Another example of this kind are sheaves on a Hausdorff space with compact support. Every partial topos E can be completed into a topos E_c by taking the inverse limit (in the appropriate 2-categorical sense) of the diagram of logical morphism i : V >--> U |---> i^* : E/U --> E/V for subterminals U,V. Coming back to the original question in case you have a big site C then consider it as living in a sufficiently big Grothendieck universe UU and take [C^op-->UU] as the presheaf topos over C. Of course, this only works if you are not afraid of big collections. But if one accepts ZF(C) why shouldn't one one spend sufficiently many Grothendieck universes as well. That such generosity is comparatively harmless when working in constructive set theories has been shown recently in Crosilla,L.;Rathjen,M. Inaccessible set axioms may have little consistency strength (Annals of Pure and Applied Logic, Volume: 115, Issue: 1-3, June 15, 2002, pp. 33-70) However, such set theories are not impredicative and the ensuing categories of presheaves are rather ``predicative toposes'' as studied by Moerdijk and Palmgren over the last few years. Thomas Streicher 20-Jun-2002 12:20:00 -0300,1785;000000000001-00000000