RE : categories: size_question_encore

["Joyal, André" <joyal.andre@uqam.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



-------- 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.


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]