Re: Small is beautiful

Re: Small is beautiful

[F William Lawvere]

Bob Pare' made the excellent point that not only size but quality 
is relevant.
I definitely agree with the spirit of his remarks. 

Bob happens to have used in passing the term 'syntactic'. 
For clarity, the use of that term needs to be sharpened to avoid
misunderstanding.
 
Actually, the term 'syntax' refers NOT to small categories such as
algebraic theories or rings, but rather to their PRESENTATION by 
signatures or by polynomial generators, et cetera. The process 
of presentation is an adjoint pair quite distinct from the 
semantical adjoint pair: both adjoint pairs have a category of 
theories or of rings in common but are otherwise quite
independent. 

In particular, syntax is NOT the adjoint of semantics. Cratylus,
Chomsky, and their 21st century followers can be refuted by
looking soberly at the actual practice of mathematics (wherein
the construction of sequences of words and of diagrams
is pursued with great care for the purpose of communication.
That syntax is only remotely dependent on the structure of the
content that is to be communicated).

Both of the functors

?--------------> theories -------------->Large categories
         Syntax                                      Semantics

are needed.  The domain category of the first can be chosen 
in various useful ways: sketches or diagrams of signatures et cetera. 

Happy new year!
 
Bill


On Fri 01/01/10  9:48 AM , pare@mathstat.dal.ca (Robert Pare) sent:
> 
> 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).
> Subfunctorsof 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
> identitieswere 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
> inpractice. Large ones which are categories of structures, corresponding
> tovarious 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
> categorieswere to be thought of as constructions of one structure from another,
> e.g.the group ring. Functors between small categories were interpretations
> ofone theory in another or reindexing or rearranging. Functors from small
> to large categories were models or diagrams in the large one. These
> kindsof functors are perhaps the most important of the four, although this
> maybe 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

[Zinovy Diskin]

In addition, a key property of semantic categories (as opposed to
syntactic) is their concreteness, i.e., their objects have carriers.
Commutativity with the forgetful functor is essential for results
stating equivalence of syntactic and semantic constructs.

Zinovy

On Tue, Jan 5, 2010 at 12:31 PM, F William Lawvere <wlawvere@buffalo.edu> wrote:
>
> Bob Pare' made the excellent point that not only size but quality
> is relevant.
> I definitely agree with the spirit of his remarks.
>
> Bob happens to have used in passing the term 'syntactic'.
> For clarity, the use of that term needs to be sharpened to avoid
> misunderstanding.
>
> Actually, the term 'syntax' refers NOT to small categories such as
> algebraic theories or rings, but rather to their PRESENTATION by
> signatures or by polynomial generators, et cetera. The process
> of presentation is an adjoint pair quite distinct from the
> semantical adjoint pair: both adjoint pairs have a category of
> theories or of rings in common but are otherwise quite
> independent.
>
> In particular, syntax is NOT the adjoint of semantics. Cratylus,
> Chomsky, and their 21st century followers can be refuted by
> looking soberly at the actual practice of mathematics (wherein
> the construction of sequences of words and of diagrams
> is pursued with great care for the purpose of communication.
> That syntax is only remotely dependent on the structure of the
> content that is to be communicated).
>
> Both of the functors
>
> ?--------------> theories -------------->Large categories
>         Syntax                                      Semantics
>
> are needed.  The domain category of the first can be chosen
> in various useful ways: sketches or diagrams of signatures et cetera.
>
> Happy new year!
>
> Bill
>

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Re: Small is beautiful

[burroni]

Dear categorists,

The question for me is not : small or large categories, but small or  
large structures.

The Bourbaki's time was the time of small structures (groups, monoïds,  
rings, spaces, etc), the "categorical time" is the time of structures  
of structures,i.e. large structures (groupoids, complete categories,  
abelian catégories, topos, etc.).
The bridge beetwen the firsts and the seconds is the Yoneda lemma,  
that is to say the introduction of logic, thus of the only true evil  
category : the category of sets. All the other large categorical  
stuctures introduced by categorists are deduced from category Set, by  
abstraction, generalisations, constructions or restrictions. In fact  
"category theory" is an (wonderfull but) inappropriate name (here the  
word category is only an important and historical keyword): the reason  
is that a lot of data (limits, classifiant object, etc) appear as  
properties because of their universal properties (unicity up to  
isomorphism). But "category theory" is an illusion, nobody studies  
seriouly the categorical structures (I don't know any structure  
theorem on the categorical structures without additional properties).
That is my starting point of  reflexion on this subject.

I think there is a true theory of the small categories, but it is not  
yet born, if it must ever exist. This theory should be, not only for  
the 1-categories, but also for the n and omega-categories (strict or  
not possibly). And, for me, the Yoneda lemma is an important tool (but  
only a tool). Such a theory may be eventually important for the  
computer science (particularly for the formal languages).

My best wishes for the new year,

Albert burroni


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Re: Small is beautiful

[Colin McLarty]

I'm not sure I understand this

> In particular, syntax is NOT the adjoint of semantics. Cratylus,
> Chomsky, and their 21st century followers can be refuted by
> looking soberly at the actual practice of mathematics (wherein
> the construction of sequences of words and of diagrams
> is pursued with great care for the purpose of communication.
> That syntax is only remotely dependent on the structure of the
> content that is to be communicated).
>
> Both of the functors
>
> ?--------------> theories -------------->Large categories
>         Syntax                                      Semantics
>
> are needed.  The domain category of the first can be chosen
> in various useful ways: sketches or diagrams of signatures et cetera.

Do you mean that if we choose some kind of sketches for the domain
category then theories are a reflective subcategory, more or less the
'definitionally closed' sketches?   Then a presentation of a theory T
would be (up to isomorphism) any unit arrow of the adjunction with T
as codomain?

best, Colin


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Re: evil

[claudio pisani]

Isn't there something "evil" in the definition of dual category itself?

Claudio





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small is beautiful

[Paul Taylor]

I largely agree with Bob Pare's New Year's Day posting, and disagree
with those who disagree with him, although Albert Burroni is right to
generalise from categories to structures.

 > ... there were two kinds of categories in practice.  [Large and
small.]

 > 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.

 > Small categories -> equality of objects okay
 > Large categories -> equality of objects not okay
 > Small is beautiful, not evil.

I would, however, like to substitute INTERNAL for SMALL.

Tradionally, a "small" category is one with a "set" of objects and
morphisms.
However, since set theory is the problem and not the solution to the
foundations of mathematics, we can avoid this by translating "set" into
"object of a topos" and more generally a lex category  (one with
pullbacks
and a terminal object") or an arithmetic universe.

(I believe that there was a development of "locally internal" instead of
"locally small" categories by some French categorists in the 1970s  --
Burroni again maybe?.)

Having done this, I would like to rescue the word "set" from the set
theorists,
and use it to mean "object of a topos", lex category or arithmetic
universe.

Vaughan Pratt's remark that "FinSet is an essentially small category" is
entirely consistent with what Bob said.

FinSet is a LARGE category for which there is a SMALL category "finset"
(whose set of objects is N) and a WEAK EQUIVALENCE, ie a full and
faithful
functor  finset->FinSet  that is essentially surjective.   This is one
of
the "most important" functors, according to Bob.

An internal category (and more generally internal structure) of course
inherits equality from its carrier, which is by definition a set (object
of a topos or lex category).

A large category or external structure has no carrier, and therefore
no notion of equality, as Bob said.

Another way of seeing the large/small or external/internal distinction
is that a small or internal structure gives a NAME to the external one,
which we may conversely call the SEMANTICS (meaning) of the name.

So, again, I agree with Bob in identifing semantics/syntax with
large/small.

You might object that this is a syntax without an alphabet of symbols.

Again we need an internal notion, this time an INTERNAL LANGUAGE.

Unfortunately, a lot of people have muddled up the terminology here.

What *I* mean by an internal language is an internal structure of a
suitable
kind for investigating formal grammar.   For example, its set of "words"
is the internal free monoid on its set of "letters", which is why we
need
an arithmetic universe.

Regrettably, other people have used the phrase "internal language" to
mean a language that is equivalent to a structure, which I call a
PROPER LANGUAGE, where I have anglicised French "propre" or Italian
"proprio".

Given, for example, an internal CCC,
it has a proper language that is an INTERNAL SIMPLY TYPED LAMBDA
CALCULUS.
 From this we may construct the internal CATEGORY OF CONTEXTS AND
SUBSTITUTIONS,
which is the internal CLASSIFYING CATEGORY for the lambda calculus,
in particular having an internal stucture-preserving functor
to the given internal CCC,
and this is an interval equivalence.

(I'm not going to go into whether it is a weak or a strong equivalence
--
see Section 7.6 of "Practical Foundations" for this.)

Besides the book, please see also
mathoverflow.net/questions/8731/categorical-foundations-without-set-
theory

Paul Taylor



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Re: small is beautiful

[burroni]

Dear Paul,

> (I believe that there was a development of "locally internal" instead of
> "locally small" categories by some French categorists in the 1970s  --
> Burroni again maybe?.)

I must mention that it is not me, but Jacques Penon who has worked on  
"locally internal" notions.

I would like to add (to my yesterday mail) some indications on what I  
have called "small categories theory". In fact, this theory begins to  
exist. It is, for instance, what is called the "higher dimensionnal  
words problem", but also a developpement on higher automata theory (I  
have made many talks on this subjet, but not published --- I can send  
a manuscript to anybody interested). I have for example proved that  
finite and finitary Lawvere theory are finitely presentable 2-monoïds  
(it is my motivation for introducing the notion of polygraphs,  
previously introduced by Ross Street under the name of computads). It  
is perhaps not well-known by the categorists because it is published  
in a computer sciences revue :

http://people.math.jussieu.fr/~burroni/mapage/highwordpb.pdf


Best,
Albert


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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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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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