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

[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: 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: evil

[claudio pisani]

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

Claudio





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evil

[John Baez]

Sorry, if it's not too late please post this one instead:

From: John Baez <baez@math.ucr.edu>
Date: Tue, Sep 14, 2010 at 3:33 PM
Subject: Evil
To: categories <categories@mta.ca>


David wrote:

Jean Benabou wrote:
>> Maybe my english isn't so "beautiful", but in all cases where "evil" has
>> been used, what is wrong with "wrong" instead?
>

I'm not so enamoured with the use of the word 'evil', but it seems to
> be more entrenched than perhaps it was intended, namely as a joke.
>

It's supposed to be funny, but I'm glad to see it become entrenched.

Why?

First, it has a very specific meaning.  A property of objects of some
category C is said to be "evil" if it holds for some object x of C but not
some isomorphic object y.  More generally: a property of objects of some
n-category is "evil" if it holds for some object x but not some equivalent
object y.  For details, see:

http://ncatlab.org/nlab/show/evil

Second, it captures the interesting state of affairs in category theory
where some definitions can be well-formed yet somehow "suboptimal" because
equations were used when isomorphisms should have been specified.

"Wrong" doesn't work here, since mathematicians use it in other important
ways: for example, "false", "incorrect" or "inappropriate".  "Evil" is, to
the best of my knowledge, never used in mathematics except in this one
technical sense.

If anybody finds the term "evil" upsettingly strong, I suggest "naughty" as
an alternative.

Best,
jb


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Re: evil

[David Yetter]

I have been following with some amusement the discussion of 'evil'.

I am mostly amused because long ago, I think in my grad student days
standing in Peter Freyd's office, I had made the suggestion that to
avoid overloading adjectives (normal and regular being particularly
abominable examples of the phenomenon) mathematicians should resort to
using adjectives that usually have a moral denotation.

Once one realizes that 'evil' exists not just for objects in categories,
but for 0- and 1-arrows in bicategories, 0-, 1- and 2-arrows in
tricategories, . . . it seems to me the proper attitude to take toward
'evil' (or strictness) is given by Saunders' dictum about generality:
"good general theory does not search for the maximum generality, but for
the right generality".  So a good (higher) categorical structure should
not search for the maximum weakness, but for the right weakness. (Or, if
you want, not search for the minimum 'evil', but the the right amount of
'evil'.)

For instance, it seems to me that structure of the category of framed
tangles in which arrows are ambient isotopy classes (rel boundary) of
framed tangles, which Shum's beautiful coherence theorem tells us is
monoidally equivalent to the free ribbon (née tortile) category one one
object generator is marred and made less useful (certainly for
application to knot theory and 3- and 4-manifold topology) by deciding
one should work instead with a (2,infinity)-category with one object,
framed point sets as 1-arrows, framed tangles as 2-arrows, isotopies as
3-arrows, isotopies of isotopies as 4-arrows, and so ad infinitum.

Or, maybe not, but only if one has an application for which the
(2,infinity)-category is the right level of weakness.

Best Thoughts,
David Yetter




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Re: evil

[Toby Bartels]

[Note from moderator: Several messages to categories apparently hung in a
mail system for several days. With apologies to posters, I am about to
post four from late Decemeber in what should have been their posting
order. Sorry about the delay,
Bob]

A dagger structure on a category should not really be considered evil at all.

If you have a functor F: C^op -> C and ask whether it is a dagger structure,
then this is (taken literally) an evil question; the answer is yes
iff F^2 = 1 and F is the identity on objects, both evil conditions.
More precisely, two isomorphic functors may have different answers.
(A non-evil version is to ask whether F is isomorphic to a dagger structure.)

However, it's not necessary to define a dagger-category as a category C
equipped with a functor F: C^op -> C such that F satisfies these conditions.
In lower-level language, we ask instead that C be equipped with an operation
that takes each morphism f: x -> y to a morphism f^\dag: y -> x
such that id^\dag = id, (f g)^\dag = g^\dag f^\dag, and (f^\dag)^\dag = f.
Nothing here refers to equality of objects; it can be formulated in a language
that (like FOLDS) does not have this concept.

Given a dagger structure on C, defined in this elementary way,
we can construct a functor \dag: C \to C^op that satisfies the evil property.
(Of course, it also satisfies the non-evil version of that property.)
But that is neither here nor there as to whether dagger structures are evil.

There is some new discussion on the nLab:
http://ncatlab.org/nlab/show/evil#daggers
In particular, Mike Shulman shows how to translate dagger structures
along equivalences of categories, proving that they are not evil.

My previous post on this subject should probably be ignored.
While any concept ~can~ be de-evilled in the way shown there,
this does not necessarily give you the concept that you want,
and indeed it need not even preserve already non-evil concepts.
(And in this case specifically, it does not seem to be correct,
as others have already argued here.)


--Toby


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