Dear David,       Whatever is published in categories is of public domain so anyone can intervene. You have nothing to apologize for.       I am not acquainted with recent work of Dorette Pronk, but I read her 1995 Utrecht thesis in detail since I was asked to do so by her advisor. In it, she refers to my paper (Marta Bunge, "An application of descent to a classification theorem for toposes" , Math. Proc. Camb. Phil. Soc. 107 (1990) 59-79), where I prove, in Corollary 5.4 to the main Theorem 5.1, the following, which is, oin the case of groupoids, what you call the third way. It says that, if W is the class of isomorphisms classes of weak equivalences of etale complete groupoids (ECG), then W admits a calculus of right fractions, and the functor B from Gpd to Top induces an equivalence ECG[W-1] \iso [Top], where [Top} denotes the category of Grotehndieck toposes (over a base S not necessarily Sets) and isomorphism classes of geometric morphisms. So, the purpose of the third way, in my view, is to prove classification theorems. However, I am not au courant of more recent developments.       As for the other two approaches I mentioned in my correspondence with Andre Joyal, their equivalence is not that obvious. In the 1-dimensional case, this is done in Bunge-Pare (1979) Proposition 2.7, and in the 2-dimensional case it is done in Bunge-Hermida (2010) Theorem 4-9.       Concerning size matters, let me observe that my construction of the stack completion (Bunge, Cahiers 1979) is meaningful regardless of size questions, that is, for any base topos S. The outcome, however, of applying it to an internal category need no longer be internal. For this reason I introduce an "axiom of stack completions" which guarantees that stack completions of internal categories be again internal,and which is satisfied by any S a Grothehdieck topos. The question of stating such an axiom as an additional axiom to the ones for elementary toposes was proposed as a problem by Lawvere in his Montreal lectures in 1974.        Good luck with your projects. Marta

Date: Mon, 11 Jul 2011 15:06:38 +0930 From: david.roberts@adelaide.edu.au To: joyal.andre@uqam.ca; martabunge@hotmail.com CC: categories@mta.ca Subject: stacks (was: size_question_encore)

Dear Marta, André, and others,

this is perhaps a bit cheeky, because I am writing this in reply to Marta's email to André, quoted below. To me it almost feels like reading anothers' mail; please forgive the stretch of etiquette.

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Marta raised an interesting point that stacks can be described in (at least) two ways: via a model structure and via descent. The former implicitly (in the case of topoi: take all epis) or explicitly needs a pretopology on the base category in question. This is to express the notion of essential surjectivity.

However, I would advertise a third way, and that is to localise the (or a!) 2-category of categories internal to the base directly, rather than using a model category, which is a tool (among other things) to localise the 1-category of internal categories. Dorette Pronk proved a few special cases of this in her 1996 paper discussing bicategorical localisations, namely algebraic, differentiable and topological stacks, all of a fixed sort.

By this I mean she took a static definition of said stacks, rather than working with a variable notion of cover, as one finds, for example in algebraic geometry: Artin stacks, Deligne-Mumford stacks, orbifolds etc, or as in Behrang Noohi's 'Foundations of topological stacks', where one can have a variable class of 'local fibrations', which control the behaviour of the fibres of source/target maps of a presenting groupoid.

With enough structure on the base site (say, existence and stability under pullback of reflexive coequalisers), then one can define (in roughly historical order, as far as I know):

representable internal distributors/profunctors = meriedric morphisms (generalising Pradines) = Hilsum-Skandalis morphisms = (internal) saturated anafunctors = (incorrectly) Morita morphisms = right principal bibundles/bitorsors

and then (it is at morally true that) the 2-category of stacks of groupoids is equivalent to the bicategory with objects internal groupoids and 1-arrows the above maps (which have gathered an interesting collection of names), both of which are a localisation of the same 2-category at the 'weak equivalences'.

Without existence of reflexive coequalisers (say for example when working in type-theoretic foundations), then one can consider ordinary (as opposed to saturated) anafunctors. Whether these also present the 2-category of stacks is a (currently stalled!) project of mine. The question is a vast generalisation of this: without the 'clutching' construction associating to a Cech cocyle a actual principal bundle, is a stack really a stack of bundles, or a stack of cocycles/descent data.

The link to the other two approaches mentioned by Marta is not too obscure: the class of weak equivalences in the 2-categorical and 1-categorical approaches are the same, and if one has enough projectives (of the appropriate variety), then an internal groupoid A (say) with object of objects projective satisfies

Gpd(S)(A,B) ~~> Gpd_W(S)(A,B)

for all other objects B, and where Gpd_W(S) denotes the 2-categorical localisation of Gpd(S) at W.

One more point: Marta mentioned the need to have a generating family. While in the above approach one keeps the same objects (the internal categories/groupoids), there is a need to have a handle on the size of the hom-categories, to keep local smallness. One achieves this by demanding that for every object of the base site there is a *set* of covers for that object cofinal in all covers for that object. Then the hom-categories for the localised 2-category are essentially small.

All the best,

David

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