stacks (was: size_question_encore)

[David Roberts <david.roberts@adelaide.edu.au>]

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.

---

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

Quoting André Joyal <joyal.andre@uqam.ca>:

> dear Marta,
>
> I apologise, I had forgoten our conversation!
> My memory was never good, and it is getting worst.
>
> You wrote:
>
>  >No, I am not thinking of the analogue of Steve Lack's model
> structure since, strictly speaking,
>  >it has nothing to do with stacks. Comments to that effect (with
> which Steve agrees) are included
>  >in the Bunge-Hermida paper. It was actually a surprise to discover
> that after trying to do what
>  >you suggest and failing.
>
> I disagree with your conclusion. I looked at your paper with Hermida.
> We are not talking about the same model structure. The fibrations in
> 2Cat(S) defined
> by Steve Lack (your definition 7.1) are too weak when the topos S does
> not satify the axiom of choice.
> Equivalently, his generating set of trivial cofibrations is too small.
>
> Nobody has read my paper with Myles
> <A.Joyal, M.Tierney: Classifying spaces for sheaves of simplicial
> groupoids, JPAA, Vol 89, 1993>.
>
> Best,
> André
>

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