size_question_encore

size_question_encore

[Eduardo Dubuc]

I have now clarified (to myself at least) that there is no canonical
small category of finite sets, but a plethora of them. The canonical one
is large. With choice, they are all equivalent, without choice not.

When you work with an arbitrary base topos (assume grothendieck) "as if
it were Sets" this may arise problems as they are beautifully
illustrated in Steven Vickers mail.

In Joyal-Tierney galois theory (memoirs AMS 309) page 60, they say S_f
to be the topos of (cardinal) finite sets, which is an "internal
category" since then they take the exponential S^S_f. Now, in between
parenthesis you see the word "cardinal", which seems to indicate to
which category of finite sets (among all the NON equivalent ones) they
are referring to.

Now, it is well known the meaning of "cardinal" of a topos ?.
I imagine there are precise definitions, but I need a reference.

Now, it is often assumed that any small set of generators determine a
small set of generators with finite limits. As before, there is no
canonical small finite limit closure, thus without choice (you have to
choose one limit cone for each finite limit diagram), there is no such a
thing as "the" small finite limit closure.

Working with an arbitrary base topos, small means internal, thus without
choice it is not clear that a set of generators can be enlarged to have
a set of generators with finite limits (not even with a terminal
object). Unless you add to the topos structure (say in the hypothesis of
Giraud's Theorem) the data of canonical finite limits.

For example, in Johnstone book (the first, not the elephant) in page 18
Corollary 0.46 when he proves that there exists a site of definition
with finite limits, in the proof, it appears (between parenthesis) the
word "canonical" with no reference to its meaning. Without that word,
the corollary is false, unless you use choice. With that word, the
corollary is ambiguous, since there is no explanation for the technical
meaning of "canonical". For example, in theorem 0.45 (of which 0.46 is a
corollary), the word does not appear. A topos, is not supposed to have
canonical (whatever this means) finite limits.

e.d.


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RE : categories: size_question_encore

[Joyal, André]

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



-------- Message d'origine--------
De: Eduardo Dubuc [mailto:edubuc@dm.uba.ar]
Date: mar. 05/07/2011 19:29
À: Categories
Objet : categories: size_question_encore
 
I have now clarified (to myself at least) that there is no canonical
small category of finite sets, but a plethora of them. The canonical one
is large. With choice, they are all equivalent, without choice not.

When you work with an arbitrary base topos (assume grothendieck) "as if
it were Sets" this may arise problems as they are beautifully
illustrated in Steven Vickers mail.

In Joyal-Tierney galois theory (memoirs AMS 309) page 60, they say S_f
to be the topos of (cardinal) finite sets, which is an "internal
category" since then they take the exponential S^S_f. Now, in between
parenthesis you see the word "cardinal", which seems to indicate to
which category of finite sets (among all the NON equivalent ones) they
are referring to.

Now, it is well known the meaning of "cardinal" of a topos ?.
I imagine there are precise definitions, but I need a reference.

Now, it is often assumed that any small set of generators determine a
small set of generators with finite limits. As before, there is no
canonical small finite limit closure, thus without choice (you have to
choose one limit cone for each finite limit diagram), there is no such a
thing as "the" small finite limit closure.

Working with an arbitrary base topos, small means internal, thus without
choice it is not clear that a set of generators can be enlarged to have
a set of generators with finite limits (not even with a terminal
object). Unless you add to the topos structure (say in the hypothesis of
Giraud's Theorem) the data of canonical finite limits.

For example, in Johnstone book (the first, not the elephant) in page 18
Corollary 0.46 when he proves that there exists a site of definition
with finite limits, in the proof, it appears (between parenthesis) the
word "canonical" with no reference to its meaning. Without that word,
the corollary is false, unless you use choice. With that word, the
corollary is ambiguous, since there is no explanation for the technical
meaning of "canonical". For example, in theorem 0.45 (of which 0.46 is a
corollary), the word does not appear. A topos, is not supposed to have
canonical (whatever this means) finite limits.

e.d.


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RE : size_question_encore

[Marta Bunge]

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
> 

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Re: size_question_encore

[Michael Shulman]

Even in a category of sets, I don't see why choice is necessary in
order to complete a small subcategory under finite limits and obtain a
small subcategory.  It seems to me that what is needed is rather the
axiom of collection, which implies that we can find some *set* of
objects containing *at least one* limit for every finite diagram in
the original small subcategory; and then we can iterate countably many
times to obtain a small category which contains at least one limit for
any finite diagram therein.  There is of course no canonical result,
and the various results obtained will not necessarily be strongly
equivalent, but it seems to me that they should all be weakly
equivalent.

And it also seems to me that the same approach should work internal to
any topos.  Collection is true internally to any topos (essentially by
the internal definition of "indexed family"), so it should still be
possible to enlarge a small internal site of definition to one that
has finite limits.  Unless there is some other subtlety that I'm not
seeing.

Mike

On Tue, Jul 5, 2011 at 4:29 PM, Eduardo Dubuc <edubuc@dm.uba.ar> wrote:
> I have now clarified (to myself at least) that there is no canonical
> small category of finite sets, but a plethora of them. The canonical one
> is large. With choice, they are all equivalent, without choice not.
>
> When you work with an arbitrary base topos (assume grothendieck) "as if
> it were Sets" this may arise problems as they are beautifully
> illustrated in Steven Vickers mail.
>
> In Joyal-Tierney galois theory (memoirs AMS 309) page 60, they say S_f
> to be the topos of (cardinal) finite sets, which is an "internal
> category" since then they take the exponential S^S_f. Now, in between
> parenthesis you see the word "cardinal", which seems to indicate to
> which category of finite sets (among all the NON equivalent ones) they
> are referring to.
>
> Now, it is well known the meaning of "cardinal" of a topos ?.
> I imagine there are precise definitions, but I need a reference.
>
> Now, it is often assumed that any small set of generators determine a
> small set of generators with finite limits. As before, there is no
> canonical small finite limit closure, thus without choice (you have to
> choose one limit cone for each finite limit diagram), there is no such a
> thing as "the" small finite limit closure.
>
> Working with an arbitrary base topos, small means internal, thus without
> choice it is not clear that a set of generators can be enlarged to have
> a set of generators with finite limits (not even with a terminal
> object). Unless you add to the topos structure (say in the hypothesis of
> Giraud's Theorem) the data of canonical finite limits.
>
> For example, in Johnstone book (the first, not the elephant) in page 18
> Corollary 0.46 when he proves that there exists a site of definition
> with finite limits, in the proof, it appears (between parenthesis) the
> word "canonical" with no reference to its meaning. Without that word,
> the corollary is false, unless you use choice. With that word, the
> corollary is ambiguous, since there is no explanation for the technical
> meaning of "canonical". For example, in theorem 0.45 (of which 0.46 is a
> corollary), the word does not appear. A topos, is not supposed to have
> canonical (whatever this means) finite limits.
>
> e.d.
>


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Re: size_question_encore

[Toby Bartels]

Mike Shulman wrote in part:

>Even in a category of sets, I don't see why choice is necessary in
>order to complete a small subcategory under finite limits and obtain a
>small subcategory.  It seems to me that what is needed is rather the
>axiom of collection, which implies that we can find some *set* of
>objects containing *at least one* limit for every finite diagram in
>the original small subcategory; and then we can iterate countably many
>times to obtain a small category which contains at least one limit for
>any finite diagram therein.

The axiom of collection guarantees only *some* appropriate set of objects,
so you need to choose one.  To iterate this countably many times,
you might need dependent choice.

Unless I'm not understanding what you're doing.


--Toby


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Re: size_question_encore

[Michael Shulman]

On Wed, Jul 13, 2011 at 9:10 PM, Toby Bartels
<categories@tobybartels.name> wrote:
>>the axiom of collection, which implies that we can find some *set* of
>>objects containing *at least one* limit for every finite diagram in
>>the original small subcategory; and then we can iterate countably many
>>times to obtain a small category which contains at least one limit for
>>any finite diagram therein.
>
> The axiom of collection guarantees only *some* appropriate set of objects,
> so you need to choose one.  To iterate this countably many times,
> you might need dependent choice.

That's a good point.  However, I think we can get around it as
follows.  We can make finitely many choices without any axiom of
choice.  Thus, for any natural number n, by applying collection n
times, we can find *some* n^th iterate of the "construction".
(Formally, we prove this by induction on n.)  Applying the axiom of
collection again over the natural numbers, we obtain a set which
contains at least one n^th iterate of the "construction" for every
natural number n.  Taking the union of this set, we should obtain a
set of objects whose corresponding full subcategory contains at least
one limit of every finite diagram therein.

Mike


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Re: size_question_encore

[Toby Bartels]

Michael Shulman wrote in part:
>We can make finitely many choices without any axiom of
>choice.  Thus, for any natural number n, by applying collection n
>times, we can find *some* n^th iterate of the "construction".
>(Formally, we prove this by induction on n.)  Applying the axiom of
>collection again over the natural numbers, we obtain a set which
>contains at least one n^th iterate of the "construction" for every
>natural number n.  Taking the union of this set, we should obtain a
>set of objects whose corresponding full subcategory contains at least
>one limit of every finite diagram therein.

OK, I buy that.  The part where we take the union
is the step that doesn't generalise to arbitrary applications of DC.


--Toby


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RE: RE: size_question_encore

[André Joyal]

Dear Marta,

You wrote:

  >What we wanted (though we did not need it) was a model structure  
similar to the Joyal-Terney but in dimension 2,
  >and of course it would be different from this one. I am sure too  
that it exists.

I am glad we agree.

Best, -Andre

-------- Message d'origine--------
De: Marta Bunge [mailto:martabunge@hotmail.com]
Date: dim. 10/07/2011 14:26
À: Joyal, André; categories@mta.ca
Objet : RE: categories: RE: size_question_encore


Dear Andre,
An addendum to my previous is in order. In Section 7 of Bunge-Hermida  
we actually discuss Lack's model structure on 2-Cat(S). We did this  
only to show that it is not suitable to get 2-stack completions, in  
case someone would think of this idea. The (ELP) is too restrictive  
for that. This may have led to your misunderstanding when we actually  
agree. What we wanted (though we did not need it) was a model  
structure similar to the Joyal-Terney but in dimension 2, and of  
course it would be different from this one. I am sure too that it  
exists.
Best,Marta



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