Small is beautiful

Small is beautiful

[Robert Pare]

I would like to add a few thoughts to the "evil" discussion.

My 30+ years involvement with indexed categories have led me
to the following understanding. There are two kinds of categories,
small and large (surprise!). But the difference is not mainly one
of size. Rather it's how well we can pin down the objects. The
distinction between sets and classes is often thought of in terms
of size but Russell's problem with the set of all sets was not one of
size but rather of the nature of sets. Once you think you have the set
of all sets, you can construct another set which you had missed.
I.e. the notion is changing, slippery. There are set theories where
you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.)
Smallness is more a question of representability: a functor may fail to
be representable because it's too big (no solution set) or, more often,
because it's badly behaved (doesn't preserve products, say). Subfunctors
of representables are not usually representable.

In our work on indexed categories, Schumacher and I had tried to treat
this question by considering categories equipped with a groupoid of
isomorphisms, which we called *canonical*, and then consider functors
defined up to canonical isomorphism. In small categories only identities
were canonical whereas in large categories, all isomorphisms were canonical.
Our ideas were a bit naive and not well developed and earned us some ridicule,
so we quietly stopped talking about it. Recently, Makkai developed
an extensive theory of functors defined up to isomorphisms, FOLDS, but
did not consider the possibility of specifying which isomorphisms ahead
of time, so small categories were not included.

When I used to teach category theory, before Dalhousie made me chuck my
chalk chuck, I would tell students there were two kinds of categories in
practice. Large ones which are categories of structures, corresponding to
various branches of mathematics we wished to study. These categories
supported various universal constructions, all defined up to isomorphism.
Two large categories are considered to be the same if they are equivalent.
It was considered impolite to ask if two objects were equal. Then
there are the small categories which are used to study the large ones.
These are syntactic in nature. For these, one can't expect the kinds of
universal constructions that large categories have, but now it's okay,
even necessary, to consider equality between objects. I went on to say
that there were then four kinds of functors. Functors between large categories
were to be thought of as constructions of one structure from another, e.g.
the group ring. Functors between small categories were interpretations of
one theory in another or reindexing or rearranging. Functors from small
to large categories were models or diagrams in the large one. These kinds
of functors are perhaps the most important of the four, although this may
be debatable. The fourth kind, from large to small are rarer. They can
be thought of as gradings or partitions of the large category.

Well, after these ramblings, perhaps my message is lost. So here it is:
Small categories -> equality of objects okay
Large categories -> equality of objects not okay
Small is beautiful, not evil.

Bob


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Re: Small is beautiful

[Vaughan Pratt]

Robert Pare wrote:
> Then
> there are the small categories which are used to study the large ones.
> These are syntactic in nature.

Don't get me started.  Oops, too late.

> For these, one can't expect the kinds of
> universal constructions that large categories have,

Not following.  FinSet is an essentially small category, what do you
mean that it doesn't enjoy universal constructions?  It's even a topos.

Then there are the categories enriched in small categories, again
subject to cardinality restrictions, which too are perfectly capable of
enjoying universal constructions.

> but now it's okay,
> even necessary, to consider equality between objects.

For small as opposed to essentially small categories, yes in some cases.
    But consider the category of ordinals truncated at say beth_2,
certainly a small category when the morphisms are the inequalities.  Are
you comfortable defining equality on the objects of this category?  (PTJ
would correctly accuse me of being inconsistent on this point.)

> Well, after these ramblings, perhaps my message is lost. So here it is:
> Small categories -> equality of objects okay
> Large categories -> equality of objects not okay

I hate to seem argumentative, Bob, but this can't possibly be the
difference between small and large.

> Small is beautiful, not evil.

Agreed, so long as this is not at the expense of large.  Nice to be able
to close on a note of consensus.  :)

Vaughan

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Re: Small is beautiful

[Eduardo J. Dubuc]

I find Bob Pare posting on large versus small super interesting, and the first
contribution since Russell (at the origin of Grothendieck Universes) with
really new and radical considerations.

Of course, Bob's posting is rather misterious, makes you think, but it is
impossible to analyze technicaly. It will be impossible also to explain it
more by writting. Needs personal disscussion.

Bob, what do you mean by "this and that ?", after the answer: Well, then it is
so !! ... but still do not understand what you really say ..., and etc etc ...

Cheers  Bob   Eduardo.


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Small2

[Robert Pare]

Thanks to all who replied to my posting either privately or on Categories.

I'd like to clarify a few points. Ross's interpretation of what I meant,
and which Vaughan agrees with, is not what I was trying to say. It is part
of the picture but not the one I was promoting.

Small categories play a different role in category theory than large
categories. Small categories are used for indexing things. Large
categories consist of the things we are indexing. Okay, syntactic was
not the right word. Combinatorial might be better although that has
finiteness overtones. Strict? Well, I'll just stick with small. Small
is not the same as essentially small. As Jeff Egger pointed out, the
category of finite sets is not small. It's not even clear what this category
is. The ZFCists would say that for each set A we get a finite set {A} and
another {{A}}, and Barwise might even wonder if these last two are distinct.
But I'm digressing.

Of course every small category can be considered as a large one (perhaps
large isn't the right word either). Then two equivalent ones would be
considered "the same". I don't think that this is how the working
categorician works. Sometimes equivalent categories are "the same" and
sometimes not. An equivalence relation is a category but we would lose
something (everything?) if we identified it with equality in the quotient
set. So I'm saying that, as a matter of "categorical hygiene", we could
be a bit more explicit about what kind "equivalence" we are allowing.
And without changing mathematical practice, I'm advocating that "small"
should imply that it's okay to talk of equality of objects.

As usual it helps put things in relief to generalize them.
Consider category theory in a world based on a topos S. A small category
is a category object in S. A large category like S or Group(S) is an
indexed category, given by a pseudo-functor S^op -> Cat, or a fibration
over S if you prefer. A small category C gives a large one by homming.
But something is lost in the process.

The object functor  Cat -> Set is not a 2-functor so if you compose it
with a pseudo-functor S^op -> Cat you get an assignment that doesn't
preserve composition in any sense. So a general indexed category doesn't
have a discrete category of objects. What you can do is take the groupoid
of isomorphisms 2-functor Cat -> Gpd, so that a large has a groupoid of
objects, thinking of a groupoid as a set with a 2-equality on it. An
indexed category gotten from a small category gives an actual functor
S^op -> Cat, so now you can compose with Ob : Cat -> Set so that a
small category has a discrete groupoid of objects.

And that's where the idea of considering categories with a specified
"equality groupoid" of isomorphisms, and equivalences defined to have
the isomorphisms from the corresponding groupoids. Small categories
have only identities, so equivalence means isomorphism, and large
categories have all isos. At the time there were no good examples
of intermediate groupoids (just products of small and large categories
and the like) but Hilbert spaces with unitary isos is a good one.

Well this is my second posting in a week bringing my life-time total
to three! Seems a good place to stop.

Best wishes for the new year. May your happiness be large and your
disappointments small.

Bob



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Re: Small is beautiful

[Ross Street]

On 03/01/2010, at 6:57 PM, Vaughan Pratt wrote:

>> For these, one can't expect the kinds of
>> universal constructions that large categories have,
>
> Not following.  FinSet is an essentially small category, what do you
> mean that it doesn't enjoy universal constructions?  It's even a  
> topos.

Dear Vaughan

Part of what Bob Paré was arguing, I believe, was that we should be  
flexible
(pun intended) about what "small" means. If "small" means "finite"  
then FinSet
is not "essentially small". Also, "small" could mean "no more than one  
element".

Ross

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Re: Small is beautiful

[Vaughan Pratt]

Ross Street wrote:
> Part of what Bob Paré was arguing, I believe, was that we should be 
> flexible
> (pun intended) about what "small" means. If "small" means "finite" then 
> FinSet
> is not "essentially small". Also, "small" could mean "no more than one 
> element".

Thanks, Ross.  Hopefully Bob will phrase it that way next time.  ;)

If 2 is the usual symmetric monoidal closed category with objects 0 and 
1 and only non-identity morphism 0 --> 1, then Chu(2,1) has four objects 
while Chu(2,0) has only three, but both are self-dual.  The CEO of 
search engine company Cuil (Old Irish for knowledge) had finite 
categories of this kind in her 1997 Ph.D. thesis.

What got me started on my previous message was that Bob was calling 
these "syntactic" when to me they were semantic.  If by "syntactic" he 
meant "finite," or more generally less than some specified ordinal, then 
I have no problem with that, other than that I'd prefer he be specific 
about the ordinal rather than vaguely saying "syntactic."

Vaughan

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re: Small is beautiful

[John Power]

Dear Colleagues,

I have not quite absorbed all the email on this yet, so may be  
repeating something already said. But perhaps it would be helpful to  
mention that, in regard to questions like this, I have found enriched  
categories helpful:

consider either

1 the functor category [->,Set] (an object is a pair of sets X and Y  
and a function from X to Y)

or

2 the category Sub(Set) (an object is a set X together with a subset  
X', and a map from (X,X') to (Y,Y') is a function from X to Y for  
which the image of X' lies in Y'

These categories, especially the first, both have the properties one  
typically seeks for a V in studying V-categories.

Spelling out what a V-category is in the second case yields a category  
C with a subcategory for which the inclusion is the identity on objects.

Happy New Year to all,

John.


Quoting Robert Pare <pare@mathstat.dal.ca>:

>
> I would like to add a few thoughts to the "evil" discussion.
>
> My 30+ years involvement with indexed categories have led me
> to the following understanding. There are two kinds of categories,
> small and large (surprise!). But the difference is not mainly one
> of size. Rather it's how well we can pin down the objects. The
> distinction between sets and classes is often thought of in terms
> of size but Russell's problem with the set of all sets was not one of
> size but rather of the nature of sets. Once you think you have the set
> of all sets, you can construct another set which you had missed.
> I.e. the notion is changing, slippery. There are set theories where
> you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.)
> Smallness is more a question of representability: a functor may fail to
> be representable because it's too big (no solution set) or, more often,
> because it's badly behaved (doesn't preserve products, say). Subfunctors
> of representables are not usually representable.

...


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Re: Small is beautiful

[Thomas Streicher]

A few little comments on "small is beautiful"

1) The nice thing about small cats is that externalizing them gives rise to
a split fibration and that allows one to speak about equality of objects.
But if we have got a split fibration P : XX -> BB it may be considered as
a small cat over \widehat{BB} = Set^{BB^op} (for Set big enough). To my
knowledge this observation is due to Jean B'enabou and also found its way
into Bart Jacob's book.

2) Identifying small with representable seems to be an idea going back to
Grothendieck already (as told to me by Jean B'enabou and taken up by him).
Namely Grothendieck's notion of representable morphism in \widehat{BB} captures
the notion of a family of small things indexed by a possibly large index object
(arbitrary presheaf). The identification of small as representable lies at the
heart of B'enabou's definitions of the properties locally small and well powered
for fibrations.
In this sense the definition of n elementary topos also amounts to a smallness
condition: namely as a category EE with finite limits such that its fundamental
fibration EE^2 -> E is well powered.

3) Use of the idea "small is representable" has been made in
Algebraic Set Theory (AST) in the formulation of Awodey, Simpson
and collaborators (see www.phil.cmu.edu/projects/ast/ for more information).
There starting from a topos EE or (when working "predicatively") from a locally
cartesian closed pretopos EE one considers the topos Sh(EE) of sheaves over EE
w.r.t. the coherent, i.e. finite cover topology. Sh(EE) is thought of a
category of classes and the full subcat of representables as the full subcat
of sets. But Sh(EE) is a bit too large because for objects X in Sh(EE) the
diagonal \delta_X (equality on X) need not be a representable morphism (and
well behaved predicates should be since otherwise separation would lead out
of sets). Thus instead of Sh(EE) one considers the full subcategory Idl(EE)
of Sh(EE) on those separated objects, i.e. those where the diagonal is a
representable mono). Notice that separated for a presheaf over EE (a split
discrete fibration over EE) means that equality is definable in the sense
of B'enabou. It was suggested to Awodey et.al. by Joyal that the separated
objects in Sh(EE) can be characterized as those presheaves over EE which can
be obtained as an "ideal colimit" of representable objects ("ideal" meaning
directed diagram of monos). A further nice characterization of X being in
Idl(EE) is that the image of a map y(A) -> X (taken in Sh(EE)) is again
representable.
Now working in Idl(EE) one can define for X in Sh(EE) its "class of subsets"
P(X) as follows: P(X)(Y) is the collection of subobjects of y(I) x X whose
source is representable, i.e. monos of the form y(J) >--> y(I) x X.
By iterating P one obtains fixpoints (not representable) of P which serve as
universes for interpreting appropriately weak set theories.

As already mentioned by Bob the set theorist Vopenka wrote a lot about
set theories where subclasses of sets needn't be sets again. He called
"semiset" a subclass of a set which is not a set itself. Although Vopenka
doesn't emphasize this point he is working in an ultra power extension of
V_\omega (because he wants the negation of the Infinity axiom to hold) and
there subclasses of a set need not be in the ultra power extension. As I have
heard (and seen some notes of talks by him) B'enabou quite some time ago worked
on a Nonstandard Theory of Classes which relates to Nelson's Internal Set Theory
like GBN to ZFC. I am vaguely aware of extensive work by NSA people on
nonstandard class and set theories motivated by similar ideas (but this was
much later) but they have a somewhat richer ontology. There are 2 books to
mention in this context

Anatoly G. Kusraev, E. I. Gordon, S. S. Kutateladze
"Infinitesimal Analysis"
Kluwer Academic Pub (2002)

V. Kanovei, M. Reeken
"Nonstandard Analysis, Axiomatically"
Springer 2004

Both views have in common that "set" has nothing to do with size but rather
with "being definable in a reasonable sense" (the collection of standard
elements of a set is typically not a set because "standard" is not a clear
cut notion). This was concealed by early axiomatizations of class theory.

I wonder now whether these two notions of "smallness" (better called "sethood")
can be reconciled more precisely.

-- Thomas


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Re: Small is beautiful

[Steve Vickers]

Dear Bob,

This reminds me of a distinction that arises topologically. In a
discrete space, equality is "OK" in the the sense that the diagonal is
an open subspace of the square. This works fine also for point-free
spaces, by a result in the Joyal--Tierney monograph. (A space X is
discrete iff all finite diagonals X -> X^n are open maps.)

That suggests working more generally with (point-free) topological
categories: the collections of objects and morphisms are spaces. Then
the ones with object equality OK are the small ones, where the spaces of
objects and morphisms are discrete, i.e. sets.

At first sight this doesn't help us with large categories. But actually
we go a long way if we generalize spaces a la Grothendieck. For example,
the "topologized version of the class of sets" is then the object
classifier S[U], a Grothendieck topos whose points are sets. That may
look like a clumsy way of replacing something unbeautiful (the large
category of sets) by something even worse. But in fact S[U] can be
presented in a way that doesn't presuppose knowledge of all of Set, by
using a site on a small category of finite sets, whose objects are the
natural numbers and whose morphisms correspond to functions between the
finite cardinals.

Many large categories, including categories of structures such as Group,
Ring etc., can be replaced in this way by topical categories, whose
collections of objects and morphisms are toposes and whose domain,
codomain etc. functors are geometric morphisms. Topical functors, again,
are made from geometric morphisms, which imposes continuity conditions
on the functors (e.g. preservation of filtered colimits, analogous to
Scott continuity).

In effect this revises the notion of "class", replacing formulae in set
theory by theories in geometric logic.

Example: In the topical category of groups, the (generalized) space of
objects is the group classifier S[Gp] while the space of morphisms is
S[GpHom], the classifier for pairs of groups with homomorphism between
them. Similarly for rings we have S[Rg] and S[RgHom]. The group ring
construction is then given by geometric morphisms S[Gp] -> S[Rg] and
S[GpHom] -> S[RgHom], satisfying the functoriality conditions.
(Actually, in this example the morphism parts are given canonically once
we have S[Gp] -> S[Rg].)

Best wishes,

Steve Vickers.

Robert Pare wrote:
> ...
> Well, after these ramblings, perhaps my message is lost. So here it is:
> Small categories -> equality of objects okay
> Large categories -> equality of objects not okay
> Small is beautiful, not evil.
>
> Bob


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Re: small2

[Marta Bunge]

This is in reply to a posting by Bob Pare. 

Dear Bob: 
You wrote: 
“As usual it helps put things in relief to generalize them. Consider category theory in a world based on a topos S. A small category is a category object in S. A large category like S or Group(S) is an indexed category, given by a pseudo-functor S^op -> Cat, or a fibration over S if you prefer. A small category C gives a large one by homming. But something is lost in the process.” 
That is precisely how I always thought of small versus large relative to a topos S (the universe of discourse). Further, a large category A (fibered over S) "is" small if and only if the corresponding pseudo-functor A: S^op--> Cat is representable. As you say, a small category C (internal to the topos S) may always be considered as a large one (via its externalization [C]: S^op ->Cat). 
These considerations came up in my and our joint work on stacks (Cahiers, 1979), and more recently in my work with Claudio Hermida on 2-stacks. The notions of a stack and of a 2-stack are taken  to be intrinsic to a topos S, that is, relative to the topology of its (regular) epimorphisms, as introduced by Lawvere 1974. 
Dimension 1. A small category C (internal to a topos S) always has a stack completion when regarded as a large category via its externalization [C]. The stack completion of C is given by yon: [C] -> [C]*= LocRep(S^(C^op)), a weak equivalence functor. Applied to a groupoid G, this gives the classification theorem for G-torsors (Diaconescu 1975). An axiom of stack completions (ASC) in its rough form says that S satisfies it if for ever small category C in S, the fibration LocRep(S^{C^op}) is representable by a category C* so that [C]* and [C*] are equivalent as fibrations. As shown by Joyal and Tierney (1991) by means of Quillen model structures, but also by a general argument involving the existence of a set of generators, (Duskin 1980), Grothendieck toposes satisfy (ASC). 
Dimension 2. The 2-dimensional analogue of the above set-up was discussed in my lecture at CT 2008 (joint work with Claudio Hermida). Our main result is that, for a topos S satisfying (ASC), any 2-category 1-stack C in S, regarded as a 2-fibration, has a 2-stack completion, to wit yon: [C] -> [C]*=LocRep(Stack^(C^op)), a weak 2-equivalence 2-functor. Applied to a 2-gerbe G and suitably interpreted, this gives a classification theorem for G-2-torsors. The validity of an appropriately formulated (ASC)^2 for a Grothendieck topos S (see slides for my lecture at CT 2008) is true by a general argument involving the existence of a set of generators. What is still missing, however, is a construction of a small 2-category 1-stack C*representing the 2-fibration [C]*= LocRep(Stack^(C^op)) in the case of a Grothendieck topos S. The Quillen model structure on 2-Cat given by Lack 2002 is not suitable for this purpose. 

Dimension n. Analogue results in higher dimensions are less tractable but a pattern emerges from the passage from dimension 1 to dimension 2.  

Remark (concerning a previous posting of yours): Although using the S-indexed versions of fibrations over S is useful, just as presentations of groups are useful, the entire discussion of stacks can be carried out at the level of fibrations (ditto 2-fibrations). I fail to understand what is all the fuss about the use of S-indexed categories if taken in that spirit. Certainly not deserving ridicule!

 You also wrote: 
“Well this is my second posting in a week bringing my life-time total to three! Seems a good place to stop.”

 To me, that is a poor reason to give. This forum could certainly profit from your interventions, but I understand your qualms, as I myself quite often abstain from an urge to intervene. 

 Happy New Year to you and everyone, Marta


************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800  
Home: (514) 935-3618
marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
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