Dear Andre,I welcome your suggestion of involving stacks in order to test universality when the base topos S does not have Choice. I have been exploiting this implicitly but systematically several times since my own construction of the stack completion of a category object C in any Grothendieck topos S (Cahiers, 1979). For instance, I have used it crucially in my paper on Galois groupoids and covering morphisms (Fields, 2004), not only in distinguishing between Galois groupoids from fundamental groupoids, but also for a neat way of (well) defining the fundamental groupoid topos of a Grothendieck topos as the limit of a filtered 1-system of discrete groupoids, obtained from the naturally arising bifiltered 2-system of such by taking stack completions. This relates to the last remark you make in your posting. Concerning S_fin, it does not matter if, in constructing the object classifier, one uses its stack completion instead, since S is a stack (Bunge-Pare, Cahiers, 1979). In my opinion, stacks should be the staple food of category theory without Choice. For instance, an anafunctor (Makkai's terminology) from C to D is precisely a functor from C to the stack completion of D. More recently (Bunge-Hermida, MakkaiFest, 2011), we have carried out the 2-analogue of the 1-dimensional case along the same lines of the 1979 papers, by constructing the 2-stack completion of a 2-gerbe in "exactly the same way". Concerning this, I have a question for you. Is there a model structure on 2-Cat(S) (or 2-Gerbes(S)), for S a Grothedieck topos, whose weak equivalences are the weak 2-equivalence 2-functors, and whose fibrant objects are precisely the (strong) 2-stacks? Although not needed for our work, the question came up naturally after your paper with Myles Tierney. We could find no such construction in the literature. With best regards, Marta
Subject: categories: RE : categories: size_question_encore Date: Wed, 6 Jul 2011 21:23:36 -0400 From: joyal.andre@uqam.ca To: edubuc@dm.uba.ar; categories@mta.ca
Dear Eduardo,
I would like to join the discussion on the category of finite sets.
As you know, the natural number object in a topos can be given many characterisations. For example, it can be defined to be the free monoid on one generator. Etc
Clearly the internal category S_f of finite set in the topos Set has many equivalent descriptions. For example, it is a a category with finite coproducts freely generated by one object u. This means that for every category with finite coproducts C and every object c of C, there is a finite coproducts preserving functor F:S_f--->C together with an isomorphism a:F(u)->c, and moreover that the pair (F,a) is unique up to unique isomorphism of pairs. It folows from this description that the category of finite sets is well defined up to an equivalence of categories, with an equivalence which is unique up to unique isomorphism.
The situation is more complicated if we work in a general Grothendieck topos instead of the topos of sets. The problem arises from the fact that in a Grothendieck topos, a local equivalence may not be a global equivalence A "global" equivalence between internal categories is defined to be an equivalence in the 2-category of internal categories of this topos. A "local"equivalence is defined to be a functor which is essentially surjective and fully faithful. Every internal category C has a stack completion C--->C' which is locally equivalent to C. A local equivalence induces a global equivalences after stack completion.
Let me remark here that the stack completion can be obtained by using a Quillen model structure introduced by Tierney and myself two decades ago. More precisely, the category of small categories (internal to a Grothendieck topos) admits a model structure in which the weak equivalences are the local equivalences, and the cofibrations are the functors monic on objects. An internal category is a stack iff it is globally equivalent to a fibrant objects of this model structure.
I propose using stacks for testing the universality of categorical constructions in a topos. For example, in order to say that the category S_f of finite sets in a topos is freely generated by one object u, we may say that for every stack with finite coproducts C and every (globally defined) object c of C, there is a finite coproduct preserving functor F:S_f--->C together with an isomorphism a:F(u)->c, and moreover that the pair (F,a) is unique up to unique isomorphism of pairs. The category of finite sets so defined is not unique, but its stack completion is unique up to global equivalence.
Finally, let me observe that the local equivalences between the categories of finite sets are the 1-cells of a 2-category which is 2-filtered. It is thus a 2-ind object of the 2-category of internal categories.
I hope my observations can be useful.
Best regards, André
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
More recently (Bunge-Hermida, MakkaiFest, 2011), we have carried out
case along the same lines of the 1979 papers, by constructing the 2- stack completion of a 2-gerbe in "exactly the same way". >Concerning
Dear Marta, I thank you for your message and for drawing my attention to your work. I apologise for not having refered to it. the 2-analogue of the 1-dimensional this, I have a question for you. Is there a model structure on 2- Cat(S) (or 2-Gerbes(S)), for S a Grothedieck topos, >whose weak equivalences are the weak 2-equivalence 2-functors, and whose fibrant objects are precisely the (strong) 2-stacks? >Although not needed for our work, the question came up naturally after your paper with Myles Tierney. We could find no such >construction in the literature. I guess you are thinking of having the analog of Steve Lack's model structure but for the category of 2-categories internal to a Grothendieck topos S. That is a good question. I am not aware that this has been done (but my knowledge of the litterature is lacunary). You may also want to establish the analog of Moerdijk's model structure for the category of internal 2-groupoids. I am confident that these model structure exists. They should be closely related to a model structure on internal simplicial groupoids <A.Joyal, M.Tierney: Classifying spaces for sheaves of simplicial groupoids, JPAA, Vol 89, 1993>. And also related to the model structure on simplicial sheaves, described in my letter to Grothendieck in 1984, but unfortunately not formally published. Best regards, Andre -------- Message d'origine-------- De: Marta Bunge [mailto:martabunge@hotmail.com] Date: ven. 08/07/2011 09:00 À: Joyal, André; edubuc@dm.uba.ar; categories@mta.ca Objet : RE : categories: size_question_encore Dear Andre,I welcome your suggestion of involving stacks in order to test universality when the base topos S does not have Choice. I have been exploiting this implicitly but systematically several times since my own construction of the stack completion of a category object C in any Grothendieck topos S (Cahiers, 1979). For instance, I have used it crucially in my paper on Galois groupoids and covering morphisms (Fields, 2004), not only in distinguishing between Galois groupoids from fundamental groupoids, but also for a neat way of (well) defining the fundamental groupoid topos of a Grothendieck topos as the limit of a filtered 1-system of discrete groupoids, obtained from the naturally arising bifiltered 2-system of such by taking stack completions. This relates to the last remark you make in your posting. Concerning S_fin, it does not matter if, in constructing the object classifier, one uses its stack completion instead, since S is a stack (Bunge-Pare, Cahiers, 1979). In my opinion, stacks should be the staple food of category theory without Choice. For instance, an anafunctor (Makkai's terminology) from C to D is precisely a functor from C to the stack completion of D. More recently (Bunge-Hermida, MakkaiFest, 2011), we have carried out the 2-analogue of the 1-dimensional case along the same lines of the 1979 papers, by constructing the 2-stack completion of a 2-gerbe in "exactly the same way". Concerning this, I have a question for you. Is there a model structure on 2-Cat(S) (or 2- Gerbes(S)), for S a Grothedieck topos, whose weak equivalences are the weak 2-equivalence 2-functors, and whose fibrant objects are precisely the (strong) 2-stacks? Although not needed for our work, the question came up naturally after your paper with Myles Tierney. We could find no such construction in the literature. With best regards, Marta
Subject: categories: RE : categories: size_question_encore Date: Wed, 6 Jul 2011 21:23:36 -0400 From: joyal.andre@uqam.ca To: edubuc@dm.uba.ar; categories@mta.ca
Dear Eduardo,
I would like to join the discussion on the category of finite sets.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
dear Marta, I apologise, I had forgoten our conversation! My memory was never good, and it is getting worst. You wrote: that after trying to do what
you suggest and failing.
Subject: RE : RE : categories: size_question_encore Date: Sat, 9 Jul 2011 12:18:45 -0400 From: joyal.andre@uqam.ca To: martabunge@hotmail.com; edubuc@dm.uba.ar; categories@mta.ca
Dear Marta,
I thank you for your message and for drawing my attention to your work. I apologise for not having refered to it.
More recently (Bunge-Hermida, MakkaiFest, 2011), we have carried out the 2-analogue of the 1-dimensional case along the same lines of the 1979 papers, by constructing the 2-stack completion of a 2-gerbe in "exactly the same way". >Concerning
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 -------- Message d'origine-------- De: Marta Bunge [mailto:martabunge@hotmail.com] Date: sam. 09/07/2011 13:41 À: Joyal, André; edubuc@dm.uba.ar; categories@mta.ca Objet : RE: RE : RE : categories: size_question_encore Dear Andre, You were indeed aware of my work and that with Pare on stacks since you are one of the few we thank for useful conversations! There were two ways to define stacks and one of them was your suggestion. One could say that one is expressed directly in terms of descent and the other in terms of weak equivalences. As it turns out, both are needed in my construction of the stack completion and similarly in the 2- dimensional case. As for which method is preferable, I do not know. Whether one constructs stack completions for categories in a Grothendieck topos using the carving out from presheaf toposes (my method), or by means of a model structure (yours), one has to resort to the existence of a generating family to keep them small. 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 attach my paper with Hermida in this connection. Section 3 makes clear what happens with Lack's model structure in dimension 1, and Section 7 considers the 2- dimensional analogue, also not suitable to get the 2-stack completion. I really meant an extension of the Joyal-Tierney model structure. Thanks for pointing out Moerdijk's work, and your old one with Tierney. I will eventually look into those. No need to respond to this. Best regards, and many thanks,Marta this, I have a question for you. Is there a model structure on 2- Cat(S) (or 2-Gerbes(S)), for S a Grothedieck topos, >whose weak equivalences are the weak 2-equivalence 2-functors, and whose fibrant objects are precisely the (strong) 2-stacks? >Although not needed for our work, the question came up naturally after your paper with Myles Tierney. We could find no such >construction in the literature.
I guess you are thinking of having the analog of Steve Lack's model
structure
but for the category of 2-categories internal to a Grothendieck topos S. That is a good question. I am not aware that this has been done (but my knowledge of the litterature is lacunary). You may also want to establish the analog of Moerdijk's model structure for the category of internal 2-groupoids. I am confident that these model structure exists. They should be closely related to a model structure on internal simplicial groupoids <A.Joyal, M.Tierney: Classifying spaces for sheaves of simplicial groupoids, JPAA, Vol 89, 1993>. And also related to the model structure on simplicial sheaves, described in my letter to Grothendieck in 1984, but unfortunately not formally published.
Best regards, Andre
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
-
André Joyal -
David Roberts -
Marta Bunge