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? Thank you, Christopher Townsend, OU 20-Jun-2002 09:56:36 -0300,2065;000000000000-00000000
On Wed, 19 Jun 2002, Christopher Townsend wrote:
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?
Thank you,
Christopher Townsend, OU
Two partial answers: (a) A result due to H. Engenes (I don't have the reference to hand, but it's early 1970s): if C is a category such that each slice C/A is (equivalent to) a small category, then [C^op,Set] is a topos. Such categories C need not be small (e.g. the ordered class of ordinals, or the category of sets and injective functions), and the resulting toposes need not be locally small (equivalently, defined over Set). I suspect that there may be a converse to this result, but I've never managed to prove it. (b) It is well known (in SGA4, and many other places) that if (the underlying category of) a site (C,T) has a small subcategory D which is dense for T, then the category of T-sheaves on C is a topos. Presumably one could weaken the smallness requirement on D to the condition in (a). Again, I don't know whether there is a converse. Peter Johnstone 20-Jun-2002 09:56:36 -0300,1592;000000000000-00000000
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 category is equivalent to a small category if and only if both it and its presheaf category are locally small. For a simple proof, see [P. Freyd and R. Street, On the size of categories, Theory and Applications of Categories 1 (9) (1995) 174-178]. So the only way a presheaf category on a locally small category C can be a Grothendieck topos is for C to be equivalent to a small category. I have a very old related conjecture: if the category E of sheaves on a locally small site is locally small then E is a Grothendieck topos. Ross 20-Jun-2002 09:56:36 -0300,3138;000000000000-00000000
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
As far ar I see, the problem is only the problem of set theory. If we allow the class of all classes, we run into Russell's paradox. If we use a theory lihe ZFC with the additional axiom that every set is a member of a Grothendieck universe (or - equvivalently - for every cardinal c there exists a strongly inaccessible cardinal larger than c) then the category of all functors from some category in a universe U to the category of all sets in U is a topos in the next universe U^+. If this presheaf category or the category of sheaves for a given site in U has a skeleton in U, then this skeleton is a sheaf in U. If one tries to build up a mathematics based on topos theory without any reference to an "outside world of sets", I do not see what "large site" means. Greetings Reinhard- 22-Jun-2002 11:19:26 -0300,5457;000000000000-00000000
participants (5)
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Christopher Townsend -
Dr. P.T. Johnstone -
Reinhard.Boerger＠FernUni-Hagen.de -
Ross Street -
Thomas Streicher