subculture

subculture

[Eduardo J. Dubuc]

As evident from the subject, this personal answer to Toby Bartels is intended
to have general incumbency.

Dear Toby, thanks for this msage, i will try to explain:

Toby Bartels wrote:
  > Eduardo J. Dubuc wrote at first:
  >
  >> Dear Toby, your choice of example is very unfortunate. Mac Lane wrote that
category theory was invented to define functor, and that functor was invented
to define "natural" transformation.
  >
  > Yes, I know; that was quite deliberate.

Well, I said "unfortunate" for those that are in favor of introducing the name
"evil" (or any other name) as a definition of "not invariant under equivalence".
You see, this is because to introduce a name the property has to be important
enough and of frequent use. To sustain your case you should have given
examples of properties (or concepts) which not being very important and of
frequent use, have nevertheless an universally accepted proper name.

  > but beyond that I have no idea what upsets you,
  > and I'm not going to worry about it any more.

I appreciate that you had worried at some point, and I am glad you do not
worry any more.

I try to explain why I sounded upset with you in my last mail because it has a
general interest concerning the question of whether  we are a subculture or
part of the mainstream of mathematics.

   Recall that this was my only mail that concerns you in particular, and that
it was in response to a mail of you, and that it was that mail that I felt
upsetting.

I quote from it:

  > Shall we stop saying "natural" and say "invariant under composition"?
  > Or is that term allowed under the grandfather clause,

"the grandfather clause" is not something nice to qualify my sayings.

  > As a proud citizen of the Ghetto of Category Land,

sounds ironic and upsetting, showing that you were very upset that i consider
certain characteristics of our group proper of a ghetto, in the sense of
isolation from the world of real mathematics. Well, I do think that one of
these characteristics is the introduction of names and terminology in an
unjustified way.  Andre Joyal call it "a subculture" (well, he just said there
is a danger to become a subculture) which if you think a little, sounds better
than "ghetto", but it is as negatively strong or even worst.

I apologize to you for using that term that you had felt insulting (and I
imagine some others in the list may have felt so)

Your msage had an overall upsetting style, and I reacted accordingly.

All the best, no hard feelings from my part.  e.d.

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Re: subculture

[Ruadhai]

Dear all,

I'd just like to point out that a quick google search of "category theory
evil" gives the correct definition, from the nLab page. As long as articles
are referenced properly, this is a non-issue. Moreover, frequently people
define things in papers which remain unused outside that one article - as
long as everything is clear, there is no problem here. With regards the
original problem, that evil is a poor choice, I personally see little point
in changing a word no one would be offended by.

Ruadhaí

On 24 September 2010 16:44, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:

> As evident from the subject, this personal answer to Toby Bartels is
> intended
> to have general incumbency.
>
> Dear Toby, thanks for this msage, i will try to explain:
>
> Toby Bartels wrote:
> > Eduardo J. Dubuc wrote at first:
> >
> >> Dear Toby, your choice of example is very unfortunate. Mac Lane wrote
> that
> category theory was invented to define functor, and that functor was
> invented
> to define "natural" transformation.
> >
> > Yes, I know; that was quite deliberate.
>
> Well, I said "unfortunate" for those that are in favor of introducing the
> name
> "evil" (or any other name) as a definition of "not invariant under
> equivalence".
> You see, this is because to introduce a name the property has to be
> important
> enough and of frequent use. To sustain your case you should have given
> examples of properties (or concepts) which not being very important and of
> frequent use, have nevertheless an universally accepted proper name.
>
> > but beyond that I have no idea what upsets you,
> > and I'm not going to worry about it any more.
>
> I appreciate that you had worried at some point, and I am glad you do not
> worry any more.
>
> I try to explain why I sounded upset with you in my last mail because it
> has a
> general interest concerning the question of whether  we are a subculture or
> part of the mainstream of mathematics.
>
>  Recall that this was my only mail that concerns you in particular, and
> that
> it was in response to a mail of you, and that it was that mail that I felt
> upsetting.
>
> I quote from it:
>
> > Shall we stop saying "natural" and say "invariant under composition"?
> > Or is that term allowed under the grandfather clause,
>
> "the grandfather clause" is not something nice to qualify my sayings.
>
> > As a proud citizen of the Ghetto of Category Land,
>
> sounds ironic and upsetting, showing that you were very upset that i
> consider
> certain characteristics of our group proper of a ghetto, in the sense of
> isolation from the world of real mathematics. Well, I do think that one of
> these characteristics is the introduction of names and terminology in an
> unjustified way.  Andre Joyal call it "a subculture" (well, he just said
> there
> is a danger to become a subculture) which if you think a little, sounds
> better
> than "ghetto", but it is as negatively strong or even worst.
>
> I apologize to you for using that term that you had felt insulting (and I
> imagine some others in the list may have felt so)
>
> Your msage had an overall upsetting style, and I reacted accordingly.
>
> All the best, no hard feelings from my part.  e.d.
>

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Not invariant but good

[Joyal, André]

Dear all,

Very briefly.

Many good things in mathematics are depending on the choice 
of a representation which is not invariant under equivalences,
or under isomorphisms. Modern geometry would not exists 
without coordinate systems. This is true also of algebra
and category theory. Algebraic structures are often described by 
generators and relations. Homological algebra is using non-canonical
projective or injective resolutions. Choosing a base point may help 
computing the fundamental group of a topological space.
Choosing a triangulation may help computing the homology groups.
Invariant notions are often constructed from notions which are not.
For example, the Euler characteristic of a space
is best explaned by using a triangulation.

Another example from homotopy theory: 
the notion of homotopy pullback square in a Quillen model category is 
invariant under weak equivalences, but its definition depends on 
the notion of pullback square which is not invariant under weak equivalences!

Part of the art of mathematics is in constructing invariant notions 
from non-invariant ones. We should recognize the usefulness and
importance of the latter. Please, let us not call them "evil"!

Best,
André

PS: We should reserve the word "evil" to name things that really are.




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<P><FONT SIZE=2>Dear all,<BR>
<BR>
Very briefly.<BR>
<BR>
Many good things in mathematics are depending on the choice<BR>
of a representation which is not invariant under equivalences,<BR>
or under isomorphisms. Modern geometry would not exists<BR>
without coordinate systems. This is true also of algebra<BR>
and category theory. Algebraic structures are often described by<BR>
generators and relations. Homological algebra is using non-canonical<BR>
projective or injective resolutions. Choosing a base point may help<BR>
computing the fundamental group of a topological space.<BR>
Choosing a triangulation may help computing the homology groups.<BR>
Invariant notions are often constructed from notions which are not.<BR>
For example, the Euler characteristic of a space<BR>
is best explaned by using a triangulation.<BR>
<BR>
Another example from homotopy theory:<BR>
the notion of homotopy pullback square in a Quillen model category is<BR>
invariant under weak equivalences, but its definition depends on<BR>
the notion of pullback square which is not invariant under weak equivalences!<BR>
<BR>
Part of the art of mathematics is in constructing invariant notions<BR>
from non-invariant ones. We should recognize the usefulness and<BR>
importance of the latter. Please, let us not call them &quot;evil&quot;!<BR>
<BR>
Best,<BR>
André<BR>
<BR>
PS: We should reserve the word &quot;evil&quot; to name things that really are.<BR>
<BR>
<BR>
<BR>
<BR>
<BR>
</FONT>
</P>

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RE : categories: subculture

[Joyal, André]

Dear All,

I am displeased with the idea that
terminology is purely conventional
and that everything is acceptable.
The "evil" terminology is promoted
by a small group of peoples active in the nLab.
It does not reflect a commun usage in the 
mathematical community.

Best,
André

-------- Message d'origine--------
De: Ruadhai [mailto:ruadhai@gmail.com]
Date: ven. 24/09/2010 20:38
À: Eduardo J. Dubuc
Cc: Categories list
Objet : Re: categories: subculture
 
Dear all,

I'd just like to point out that a quick google search of "category theory
evil" gives the correct definition, from the nLab page. As long as articles
are referenced properly, this is a non-issue. Moreover, frequently people
define things in papers which remain unused outside that one article - as
long as everything is clear, there is no problem here. With regards the
original problem, that evil is a poor choice, I personally see little point
in changing a word no one would be offended by.

Ruadhaí


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Re: subculture

[David Leduc]

Ruadhai <ruadhai@gmail.com> wrote:
>  With regards the
> original problem, that evil is a poor choice, I personally see little point
> in changing a word no one would be offended by.

It is certainly not the case of the work "kosher" used by some people
on this list.


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Re: subculture

[Fred Linton]

It may well be, as Ruadhaí points out, that

> ... a quick google search of "category theory
> evil" gives the correct definition, from the nLab page.

But does a search for "evil" give you that, as well?

> ... With regards the
> original problem, that evil is a poor choice, I personally see little
> point
> in changing a word no one would be offended by.

The word "evil" is not a mere anagram for "live", "veil", or "vile",
but has a meaning of its own, replete with connotations of opprobrium
for whatever it is used as descriptive adjective for.

On that ground, I would propose, it *is* a poor choice -- unless
you see little point in respecting the ordinary meaning of the word
"evil", or great value in offending those who would respect it.

It's an excellent choice if your goal is precisely to offend those
for whom "evil" already has a meaning incompatible with its proposed
mathematical use here. Perhaps there are even better choices, though:
Why not "demented", "testicular", "terrorist", or "gay" instead? Or
some other negative word even better suited to the purpose of getting
a rise out of the literal-minded, if all you really want to do with it
is to "épater les bourgeois"?

With dumbfounded cheers, -- Fred



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Re: Not invariant but good

[John Baez]

Dear Andre -

> Many good things in mathematics are depending on the choice
> of a representation which is not invariant under equivalences,
> or under isomorphisms. Modern geometry would not exists
> without coordinate systems.

I agree.  I think you're arguing against a position that nobody
here has espoused.

A coordinate system is a structure, not a property.  In my
earlier email I said a *property* is evil if it's not invariant under
equivalences.  But I'd say a *structure* is evil if it's not *covariant*
under equivalences.  Coordinate systems are covariant under
equivalences, so they're not evil.

Let me expand on this a bit, first for properties and then for
structures.

Say we have a groupoid C.  A (possibly evil) property of objects
in C is a map

F: Ob(C) -> {F,T}

where Ob(C) is the class of objects of C and {F,T} is the set of
truth values.  I say the property is non-evil if it extends to a
functor

F: C -> {F,T}

where now we regard {F,T} as a discrete groupoid.

For example, suppose C is the groupoid of vector spaces
over your favorite field.

A typical non-evil property is "being 5-dimensional".  A typical
evil property is "having the empty set as its origin".  (In ZF set
theory, we can take any vector space and ask whether
its zero element happens to be the empty set.)

I think most mathematicians would be happy to see a theorem
that begins

Theorem: If V is a vector space that is 5-dimensional...

but somewhat surprised to see a theorem that begins:

Theorem: If V is a vector space with the empty set as its origin...

We instinctively feel that any theorem of the second sort
could have been phrased better: how could it really matter
that the origin is the empty set?

Now, structures.  Again, let C be a groupoid.  A (possibly evil)
structure on objects in C is a map

F: Ob(C) -> Set

The idea is that for any object c in C, F(c) is the set of
structures that can be put on that object.  I say the structure
is non-evil if it extends to a functor

F: C -> Set

If C is the groupoid of vector spaces, a typical non-evil
structure is a basis: here F(c) is the set of bases of the
vector space c.   A typical evil structure would be "a basis,
if the underlying set of c is the real numbers, but two bases
otherwise."

Again, I think most mathematicians would feel happy to work
with the non-evil structure, but somewhat uncomfortable
working with the evil one.  That's the feeling that this concept
of "evil" is trying to formalize.

Note that a non-evil structure on finite sets is what you call a
"species".  You wisely avoided studying the evil ones.

Best,
jb


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Re: subculture

[Eduardo J. Dubuc]

I give my opinion simultaneously to several postings;


(1) Ruadhai wrote:

  >> With regards the
  >> original problem, that evil is a poor choice, I personally see little point
  >> in changing a word no one would be offended by.

Precisely, we should not accept a terminology just because it does not offend
anybody.

Jesus Christ !!, with this philosophy we could accept any ridiculous
terminology so far "it does not offend".

Terminologies may have an strong "ideological" connotation. To call something
"evil" it is not harmless, neither unintentional (do not forget the
unconscious part of the brain of those that promote "evil").

It also has a marketing attitude (compare with "catastrophe theory" to refer
to the classification of singularities of C^oo maps).

(1) David Leduc wrote:

  > It is certainly not the case of the work "kosher" used by some people
  > on this list.

Well, I can not imagine the word "not kosher" to offend anybody if applied to
something that it is not accepted by the rules of a discipline.

(like constructivism, intuitionism, or "accept only concepts invariant by
equivalence").

Of course, if "not kosher = evil", then some people would not like it. But the
blame is in those that introduced the terminology "evil" to refer to something
which is not necessarily evil.

(3) Joyal wrote:

  > I am displeased with the idea that
  > terminology is purely conventional
  > and that everything is acceptable.
  > The "evil" terminology is promoted
  > by a small group of peoples active in the nLab.
  > It does not reflect a commun usage in the
  > mathematical community.

Well, certainly true what Andre says.

"evil" is a terminology so far used by some people, certainly not the
mathematical community. It has also the weakness to remain for ever within a
"subculture" that we do not want to be identified with category theory.

(Rene Thom had sufficiently strong contributions to mainstream mathematics to
impose his "catastrophe" terminology. This is not the case for the "evil"
terminology).

All the best to all, and welcome controversy !!.

e.d.

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Re: Not invariant but good

[Peter Selinger]

Hi John,

thanks for trying to move the discussion away from terminology and
back to actual mathematical matters. In order to focus on the math and
not on the terminology, let me today use the word "XXXX" instead of
"evil".

I don't think the notion used in your examples is general enough. For
example, fix some groupoid C, and consider the property of an object
x: "x is isomorphic to exactly 3 objects of C".  To me, this is
clearly XXXX, because it is not invariant under equivalences of
C. Yet, according to the definition you used in this email, it extends
to a functor C -> {F,T}, and therefore is non-XXXX.

For a property P of objects x of a category C, "being invariant under
isomorphisms of objects in C" is strictly weaker than "being invariant
under equivalences of C". Proof: Clearly, any isomorphism of C can be
mapped to an identity of some category C' by some equivalence of
categories.  Therefore, any property that is invariant under
equivalences of categories is invariant under isomorphisms of
objects. The above example shows that the converse is not true.

I think for XXXXness of structures, a similar refinement is needed.
To me, the intuitive concept of XXXX for structures is "cannot be
transported along equivalences such that the equivalence becomes
structure preserving". To say it more explicitly: if category C has
the structure, and category C' is equivalent to C (as a category),
then C' can be equipped with a structure in such a way that the
equivalence (both directions) is structure preserving.

This is a fairly subtle concept, not least because it depends on the
precise 2-category in question (to fix what "equivalence" and
"structure preserving" means). For example, whether the structure of
being "strictly monoidal" is XXXX or not depends on what one means by
"structure preserving" (e.g., strict monoidal or strong monoidal
functors). The natural transformations need to be specified too, so
that one can define "structure preserving equivalence".

I tried to give a more general and precise 2-categorical definition on
the categories list on January 3, 2010, but I am not sure I got it
quite right. I think it was Mark Weber who also pointed out, around
the same time, that one person's XXXX concept is another person's
non-XXXX concept - in a different 2-category.

The fact that being XXXX depends on an ambient 2-category means that
it is not a moral judgment, and people should not be offended by
it. Some perfectly useful things can be XXXX sometimes, and some
perfectly useless things can be non-XXXX. For example, even a
not-very-natural property like "there are exactly 3 objects isomorphic
to x" can be non-XXXX when viewed in the right 2-category. For
example, this is the case in the 2-category of categories, functors,
and identity natural transformations.

Last comment. Thomas Streicher brought up the example of a fibration
P: XX -> BB as a concept that was XXXX but very useful. But I don't
think this concept is actually XXXX. Certainly if one thinks of the
fibration as a *structure* on BB, then this transports very nicely
along equivalences. Namely, given any equivalence BB <--> BB', one can
find a fibration P' : XX' -> BB' which is equivalent, as a fibration,
to P. Right?

So I don't think it is correct to identify the concept of XXXX with
"having to talk about equality". Rather, it should be defined in some
2-categorical way. See also Mike Shulman's post from January 4, which
discussed this distinction in more depth.

-- Peter


John Baez wrote:
>
> Dear Andre -
>
>> Many good things in mathematics are depending on the choice
>> of a representation which is not invariant under equivalences,
>> or under isomorphisms. Modern geometry would not exists
>> without coordinate systems.
>
> I agree.  I think you're arguing against a position that nobody
> here has espoused.
>
> A coordinate system is a structure, not a property.  In my
> earlier email I said a *property* is evil if it's not invariant under
> equivalences.  But I'd say a *structure* is evil if it's not *covariant*
> under equivalences.  Coordinate systems are covariant under
> equivalences, so they're not evil.
>

...


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RE : categories: Re: Not invariant but good

[Joyal, André]

Dear John,

I agree with you that some of the examples 
in my list can be regarded as covariant structures.
But not all of them. Especially the example of
pullback squares in a model category. In fact,
the notion of fibration in a model category is
also not invariant under weak equivalences, since
every map is, up to a weak equivalence, a fibration. 
The notion of Grothendieck fibration is also not invariant 
under equivalences of categories, since the composite of a 
Grothendieck fibration with an equivalence is not a 
Grothendieck fibration in general. One could introduce 
a weaker notion of Grothendieck fibration which repairs this absence
of invariance but the usual notion of a Grothendieck 
fibration will remain important. I am reluctant to
call the notion of Grothendieck fibrations "evil".

I feel that the whole controversy about the "evil" terminology
is preventing us from discussing rationally and fruithfully
important foundational issues. The word is very negative
and polarising. Nobody likes to be told that he has
done something "evil" when he has done nothing so.

I guess you have introduced the "evil" terminology
because you wanted peoples to pay attention to
the fact that certain constructions in category theory
and higher category theory are not invariant under
equivalences. If this is so, you have succeeded in your goal.
But please, could you agree to change the terminology?

Best,
André


-------- Message d'origine--------
De: John Baez [mailto:baez@math.ucr.edu]
Date: sam. 25/09/2010 23:29
À: categories
Objet : categories: Re: Not invariant but good
 
Dear Andre -

> Many good things in mathematics are depending on the choice
> of a representation which is not invariant under equivalences,
> or under isomorphisms. Modern geometry would not exists
> without coordinate systems.

I agree.  I think you're arguing against a position that nobody
here has espoused.

A coordinate system is a structure, not a property.  In my
earlier email I said a *property* is evil if it's not invariant under
equivalences.  But I'd say a *structure* is evil if it's not *covariant*
under equivalences.  Coordinate systems are covariant under
equivalences, so they're not evil.

Let me expand on this a bit, first for properties and then for
structures.

...


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RE : Re: Not invariant but good

[John Baez]

Andre wrote:

>I guess you have introduced the "evil" terminology
>because you wanted people to pay attention to
>the fact that certain constructions in category theory
>and higher category theory are not invariant under
>equivalences.  If this is so, you have succeeded in your goal.

Good!  But I never really thought of it as a piece of "terminology"
that I "introduced".  I've never used in any published paper,
for example.  I've always thought of it as a JOKE.

In the future I'll try to avoid telling this joke in the presence of
people who find it upsetting. This should be easy, because
I'm not working on pure math anymore, except for a few
projects that I'm trying to finish up.  I'm working on
environmental issues.

The planet is headed for an ecological disaster within this
century.  We cannot prevent it, we can only minimize the
damage.  I hope some people on the category theory
mailing list can help out here:

http://www.azimuthproject.org/azimuth/show/Azimuth+Project

If you aren't sure how to help, send me an email.

Best,
jb


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Re: RE : categories: Re: Not invariant but good

[Thomas Streicher]

Thanks Andr'e for pointing out by various examples why it is not always wise
to insist on invariance under equivalence or weak equivalence. This in my
opinion is the real issue and not whether "evil" is a tasteless name or not.

The problem rather is that people using "evil" really mean it so even if they
deny it.

An little comment on structure versus property which is an importnat
distinction in my eyes. The notion of Grothendieck fibration is a property
of functors and not an additional structure. However, the notion of fibration
in a(n abstract) 2-category can be formulated only postulating a certain kind
of structure which, however, is unique up to canonical isomorphism. But this
amounts to defining Grothendieck fibrations in terms of cleavages (which
certainly are all canonically isomorphic). But choosing cleavages amounts
to accepting very strong choice principles which is maybe no real problem
but at least aesthetically moderately pleasing.

Thomas


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no joke

[Joyal, André]

Dear John,

>But I never really thought of it as a piece of "terminology"
>that I "introduced".  I've never used in any published paper,
>for example.  I've always thought of it as a JOKE.

>In the future I'll try to avoid telling this joke in the presence of
>people who find it upsetting. 

I am not offended personally.
You are free to repeat the "joke" ad nauseum, but who is laughing?
Some of your followers are taking the "joke" quite seriously:

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

You wrote:

>I'm not working on pure math anymore, except for a few
>projects that I'm trying to finish up. 

You are a good mathematician. 
You have made important contributions to higher category theory.
I am sure you can make further contributions.
I hope you will not leave math. 
Everybody likes your web pages on maths and physics.

You wrote:

>I'm working environmental issues.

I applaude to your involvement:

http://www.azimuthproject.org/azimuth/show/Azimuth+Project

I agree that the planet is presently heading for an ecological disaster.
You wrote that the situation about climate change is hopeless.
I can understand your pessimism, but how do you really know?
You also wrote that not all human beings will die.
Is this a source of optimism?
Who should be saved? Who can be abandoned?
You also wrote that we should consider using geo-engineering.
How different is your position from Bjorn Lomborg's?

http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521138567

http://climateprogress.org/2010/09/01/the-lomborg-deception/

http://www.realclimate.org/index.php/archives/2009/08/a-biased-economic-analysis-of-geoengineering/

Best,
André



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Re: Not invariant but good

[Michael Shulman]

On Tue, Sep 28, 2010 at 3:18 AM, Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
> The notion of Grothendieck fibration is a property
> of functors and not an additional structure. However, the notion of fibration
> in a(n abstract) 2-category can be formulated only postulating a certain kind
> of structure which, however, is unique up to canonical isomorphism. But this
> amounts to defining Grothendieck fibrations in terms of cleavages (which
> certainly are all canonically isomorphic). But choosing cleavages amounts
> to accepting very strong choice principles which is maybe no real problem
> but at least aesthetically moderately pleasing.

I'm not sure what you mean here.  The notion of fibration in a
2-category can be defined as a property if you like: a morphism E -->
B in a 2-category K is a fibration if all the induced functors K(X,E)
--> K(X,B) are fibrations and all commutative squares induced by
morphisms X --> X' are morphisms of fibrations (preserve cartesian
arrows).  This is equivalent to giving some structure on E --> B, but
that structure is unique up to unique isomorphism when it exists.

(One can argue that both ordinary fibrations and fibrations in a
2-category are actually "property-like structures," or "properties
that are not necessarily preserved by morphisms," since their
forgetful functors are pseudomonic but not full.  But that applies
equally to both.)

It is true that if one has a Grothendieck fibration between categories
in the ordinary sense, then it only becomes an internal fibration in
the naively defined 2-category Cat if we have the axiom of choice,
since the latter amounts to saying that we can simultaneously choose
cartesian liftings for any families of objects of E and morphisms of B
we might want to pull them back along.  (I don't think any "global
choice" is necessary, since the definition doesn't require us to make
such a choice simultaneously for every possible family--only that for
any particular family, we -could- make such a choice.)

However, in the absence of the axiom of choice, the naive definition
of "functor" is not very well-behaved; it's better to use
"anafunctors," or equivalently to invert the weak equivalences
(fully-faithful and essentially surjective functors) in the naively
defined 2-category Cat.  In the resulting bicategory, I think any
ordinary Grothendieck fibration will indeed be an internal fibration,
without any need for choice.  (Of course, "internal fibration" should
probably now be interpreted in the looser sense of Street, as
appropriate when working in a non-strict 2-category.)

Mike


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Re: no joke

[John Baez]

[Note from moderator: This is a very long post and off topic. Further
discussion of the subject must happen elsewhere. ]

To the moderator - while this post is a bit off topic, I hope you
can indulge me and post it, if only because it explains why
I am no longer working on n-categories.  The math is near the
end.

Dear Andre -

You write:

> You wrote that the situation about climate change is hopeless.
> I can understand your pessimism, but how do you really know?

First of all, climate change is already upon us.  So far this year
has been the hottest in recorded history, and we're seeing
precisely the erratic precipitation patterns that we'd expect from
global warming.  Droughts and heat waves in Russia have caused that
country -  the 3rd largest grain exporting country - to ban exports of
wheat.  Floods displaced 2 million people in Pakistan, 2 million in
Nigeria, and hundreds of thousands Uganda, Kenya and Sudan.  We're
also seeing unusual things like tornadoes in New York City, and coral
reefs dying from overheated water in Indonesia.  Etcetera.

Second of all, once carbon dioxide is in the atmosphere, a large
portion stays there for hundreds or thousands of years.  So, to
prevent the CO2 concentration from rising above 450 ppm, truly
astounding actions would be required STARTING NOW.

Let me quote Stewart Brand's summary of a talk by the engineer
Saul Griffith:

>What would it take to level off the carbon dioxide in the
>atmosphere at 450 parts per million (ppm)? That level
>supposedly would keep global warming just barely manageable
>at an increase of 2 degrees Celsius. There still would be massive
>loss of species, 100 million climate refugees, and other major
>stresses. The carbon dioxide level right now is 385 ppm, rising
>fast. Before industrialization it was 296 ppm.  America's leading >climatologist, James Hansen, says we must lower the
>carbon dioxide level to 350 ppm if we want to keep the world
>we evolved in.

>The world currently runs on about 16 terawatts (trillion watts)
>of energy, most of it burning fossil fuels. To level off at 450 ppm
>of carbon dioxide, we will have to reduce the fossil fuel burning to 3 >terawatts and produce all the rest with renewable energy, and we
>have to do it in 25 years or it's too late. Currently about half a
>terawatt comes from clean hydropower and one terawatt from clean >nuclear.  That leaves 11.5 terawatts to generate from new
>clean sources.

>That would mean the following. (Here I'm drawing on notes
>and extrapolations I've written up previously from discussion
>with Griffith):

>"Two terawatts of photovoltaic would require installing 100
>square meters of 15-percent-efficient solar cells every second,
>second after second, for the next 25 years. (That's about 1,200
>square miles of solar cells a year, times 25 equals 30,000 square
>miles of photovoltaic cells.) Two terawatts of solar thermal? If it's
>30 percent efficient all told, we'll need 50 square meters of highly >reflective mirrors every second. (Some 600 square miles a year,
>times 25.) Half a terawatt of biofuels? Something like one Olympic
>swimming pool of genetically engineered algae, installed every
>second. (About 15,250 square miles a year, times 25.) Two
>terawatts of wind? That's a 300-foot-diameter wind turbine every
>5 minutes. (Install 105,000 turbines a year in good wind locations,
>times 25.) Two terawatts of geothermal? Build three 100-megawatt
>steam turbines every day — 1,095 a year, times 25. Three terawatts
>of new nuclear? That's a 3-reactor, 3-gigawatt plant every week —
>52 a year, times 25".

[...]

>Meanwhile for individuals, to stay at the world's energy budget at
>16 terawatts, while many of the poorest in the world might raise
>their standard of living to 2,200 watts, everyone now above that
>level would have to drop down to it.

I believe actions of this scale will not happen in time, and thus
it's hopeless to prevent a disaster.  However, that does not mean
we should give up!   Even if a disaster of some sort is certain, there
are different degrees of disaster, and it’s our responsibility to
minimize the disaster.

> You also wrote that not all human beings will die.
> Is this a source of optimism?

It is for me!  I like people.

> Who should be saved? Who can be abandoned?

Luckily for me, there are lots of useful things I can do without
knowing the answer to this question.

> You also wrote that we should consider using geo-engineering.

More precisely, I think we should study geo-engineering.
People are already considering it, regardless of what I say.
The pressure to use it will become intense as things get worse.
Some forms, such as biochar, might be quite safe if managed
properly.  Others, which do not reduce the CO2 level, could be
very dangerous.

(People disagree on whether biochar counts as geo-engineering.
It simply means turning agricultural waste into charcoal and
burying it: http://www.azimuthproject.org/azimuth/show/Biochar)

> How different is your position from Bjorn Lomborg's?

I haven't read Lomborg's new book yet, so I'm not sure.
However, I get the feeling that he advocates geo-engineering
as a cost-effective way to prevent global warming.  My position
would then be different.

I believe we won't do anything significant about global warming
until it's too late and there are massive social disruptions. Then
there will be extreme pressure to try anything, including
geo-engineering.  So, I think it's important to study geo-engineering
along with all other solutions.  If people who don't like it refuse to
study it, only people who like it will study it - so they'll be the ones
that governments will consult.

Now, about mathematics:

> I hope you will not leave math.
> Everybody likes your web pages on maths and physics.

Thanks very much!

I'm trying to figure out how to combine my interest in math with my
desire to help save the planet.  There are lots of options; the
problem is finding one that achieves a significant effect.  To do pure
math, I just needed to follow the beauty.  But this is different.

There are easy things, and harder things.

Everyone who teaches math can incorporate real-world examples
in their teaching, and use them to educate people about the world
we live in.  For example: overfishing can cause fish populations to
crash. This can be seen in a very simplified way using the equation

dP/dt = kP - c

or in more realistic ways using more complicated equations.  A
mathematician friend of mine was shocked when two colleagues
of his, experts on differential equations, didn't know this.  When I
heard his story, I realized we should be talking about overfishing
every time we teach kids how to solve separable differential equations.

Another slightly more sophisticated example involves the role of
feedback in global warming.  I'm sure there are many more.  I'll
collect them and make them easy to find.

More generally, everyone who teaches math or science can help students
think clearly about real-world problems.  This is urgent!
Logic and statistics are vital.

Mathematicians and scientists should also spend more of their
research time on environmental issues. For students this
should be easy: these issues will become ever more important,
so working on them is a good road to a successful career -
much easier than, say, category theory.

Other people may feel they're too old to change directions.  But
in fact, I've found that the best way to become young is to try
something new.  Being tenured makes this very easy to do.

The hardest part is figuring out which actions will have the
optimal effect.  For me, the first step is to quit pure math,
work on environmental issues, and trying to convince large
numbers of scientists to turn their attention in that direction.
But the last part will only work if I can find a path forward that
looks attractive.

Best,
jb

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Re: Not invariant but good

[Richard Garner]

> It is true that if one has a Grothendieck fibration between categories
> in the ordinary sense, then it only becomes an internal fibration in
> the naively defined 2-category Cat if we have the axiom of choice,
> since the latter amounts to saying that we can simultaneously choose
> cartesian liftings for any families of objects of E and morphisms of B
> we might want to pull them back along.  (I don't think any "global
> choice" is necessary, since the definition doesn't require us to make
> such a choice simultaneously for every possible family--only that for
> any particular family, we -could- make such a choice.)

Though we can always make such a simultaneous choice as soon as we
have it in one particular case. Given the Grothendieck fibration p:
E->B in Cat, and letting X denote the comma object (B,p), it is enough
to choose a lifting for the X-indexed family of morphisms of B
corresponding to the projection X --> B^2 at the X-indexed family of
objects of E corresponding to the projection X --> E; for then to give
a Y-indexed family of morphisms in B and a Y-indexed family of objects
of E over their codomains is to give a morphism Y -> X, and so the
chosen lifting for the latter induces a chosen lifting for the former.

Richard


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Re: Not invariant but good

[Thomas Streicher]

Dear Mike,

> I'm not sure what you mean here.  The notion of fibration in a
> 2-category can be defined as a property if you like: a morphism E -->
> B in a 2-category K is a fibration if all the induced functors K(X,E)
> --> K(X,B) are fibrations and all commutative squares induced by
> morphisms X --> X' are morphisms of fibrations (preserve cartesian
> arrows).  This is equivalent to giving some structure on E --> B, but
> that structure is unique up to unique isomorphism when it exists.

The definition you give entails that a "generalised" fibration is actually
a Grothendieck fibration (since Cat(1,E) is isomorphic to E). This way you
don't get closure under precomposition by equivalences. I also don't see why
cartesiannness of the functors induced by X --> X' should amount to a choice
of structure (cartesiannness of a functor is a property and not additional
structure). Moreover, this requirement is a property of Grothendieck fibrations
which can be established when having strong choice available.

I was rather alluding to the notion of fibration in 2-cats as can be found
in part B of the Elephant where a fibration is defined as a 1-arrow together
with additional structure.

The definition you gave above (which is not more general) is the obvious thing
to do in case K is not wellpointed enough (as Cat is).

Moreover, your definition of fibration in a 2-category is based on Grothendieck
fibrations and thus employs equality of 1-cells. Since 1-cells are objects of
a category it should be "evil" to speak about their equality. Not that it were
a problem to me...

Thomas

PS  Your definition of fibration in a 2-cat looks much simpler than what I
could find in the papers by Street and Johnstone. That's nothing to complain
about but where is it from? It seems to me the appropriate one when generalising
from Cat to more general 2-cats.


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Re: Not invariant but good

[Thomas Streicher]

> However, in the absence of the axiom of choice, the naive definition
> of "functor" is not very well-behaved; it's better to use "anafunctors,"

Nothing against anafunctors but it is an exaggeration to say that in absence
of choice the usual notion of functor is not well-behaved. One just loses that
full and faithful and essential surjective entails equivalence. That's like
abandoning the notion of surjective map in case we can't split them all.

Thomas


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Re: Not invariant but good

[Michael Shulman]

Dear Thomas,

I did not intend to "generalize" anything, only to restate the
definition.  There are at least four equivalent definitions of
fibration in a 2-category that I know of:

- the "representable" one which I gave,
- having an adjoint one-sided inverse to a certain morphism between
comma objects
- being an algebra for a certain monad on a slice 2-category
- the one in the Elephant

All the definitions have two versions: a "strict" one a la
Grothendieck (which only makes sense in a strict 2-category) and a
"weak" one a la Street (which makes sense in any bicategory).  All the
strict notions are equivalent to each other, and all the weak notions
are equivalent to each other.  The idea of the equivalence of the
first three is essentially contained in Richard's remark, while a
proof of the equivalence of the Elephant's version with the
representable one can be found in Peter Johnstone's article
"Fibrations and partial products in a 2-category."

Weak fibrations are closed under composition with equivalences, while
strict ones are not.  In Cat and probably other well-behaved
2-categories, being a weak fibration is the same as being the
composite of a strict fibration and an equivalence, and so it ought to
surprise no one that in such cases it is sufficient to consider strict
fibrations.  It's generally only in the bicategorical world that weak
fibrations become important.

All the definitions can also be described either as "properties"
(such-and-such thing exists) or as "structure" (equipped with
such-and-such thing), since in all cases the such-and-such is unique
up to unique isomorphism when it exists.  This is the situation also
referred to as "property-like structure."  But perhaps you were
originally referring instead to the structure required on the
2-category itself?  It's true that the latter three definitions
require existence of some limits in the 2-category, while the
representable version does not.

Finally, in Cat with choice, the strict notions are all equivalent to
the usual Grothendieck fibrations, while the weak ones are equivalent
to Street's.  (Street actually originally gave his definition in a
general bicategory, and only later specialized it to Cat.)

Does this clarify what I meant?

> Nothing against anafunctors but it is an exaggeration to say that in absence
> of choice the usual notion of functor is not well-behaved.

Perhaps "not well-behaved" was a poor choice of words since it implies
a value judgement, but I think it is correct to say that in the
absence of choice, category theory becomes very unfamiliar unless we
replace functors with anafunctors.  For instance, if we insist on
using only functors, then a category with finite products does not
necessarily become a monoidal category, as there is no "product"
functor from A×A to A.  Also, without choice the 2-category Cat using
functors is not even a regular 2-category, let alone a 2-topos.  Since
the defining characteristics of Set include that it is a well-pointed
1-topos, it seems unlikely to me that one will be able to get much of
anywhere with a version of Cat that is not a 2-topos.

Mike


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Re: Not invariant but good

[Michael Shulman]

On Fri, Oct 1, 2010 at 2:24 AM, Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
> Well, but then we can work with categories with a chosen structure, e.g.
> chosen products; that's what is recommended in the Elephant and it looks to
> me as close to practice; it's very unlikely to have a nonconstructive proof
> of existence of products.

Unlikely, but it happens occasionally.  For instance, if the morphisms
of a category are equivalence classes, then a "construction" of
equalizers or pullbacks might require choosing representatives; this
actually happens at one point in the Elephant.  The "small complete
categories" in realizability topoi are also generally "weakly
complete," in the sense that "every small diagram has a limit" is true
in the internal logic, but not "strongly complete" in the sense that
there exist internal limit-assigning (non-ana) functors.  The property
of "strong completeness" is also not in general preserved or reflected
by weak equivalence functors.

One can of course develop a theory which distinguishes between weak
and strong equivalence and weak and strong completeness, but I think
it's reasonable to call it "unfamiliar" to most category theorists.
It feels to me like trying to do work constructively with topological
spaces and therefore having to talk about [0,1] being complete and
totally bounded but not compact, instead of realizing that when
working constructively, one should really replace topological spaces
with locales.  Just as in set theory, no axiom of choice is necessary
to define a function whose values are individually uniquely
determined, it seems to me that no axiom of choice should be necessary
in category theory to define a functor whose values are individually
uniquely determined up to unique isomorphism.  But obviously this is a
subjective judgement.

Mike


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Re: Not invariant but good

[Thomas Streicher]

Dear Michael,

> Does this clarify what I meant?

Yes, indeed that has been very helpful. 

> Perhaps "not well-behaved" was a poor choice of words since it implies
> a value judgement, but I think it is correct to say that in the
> absence of choice, category theory becomes very unfamiliar unless we
> replace functors with anafunctors.  For instance, if we insist on
> using only functors, then a category with finite products does not
> necessarily become a monoidal category, as there is no "product"
> functor from A×A to A.  Also, without choice the 2-category Cat using
> functors is not even a regular 2-category, let alone a 2-topos.  Since
> the defining characteristics of Set include that it is a well-pointed
> 1-topos, it seems unlikely to me that one will be able to get much of
> anywhere with a version of Cat that is not a 2-topos.

Well, but then we can work with categories with a chosen structure, e.g.
chosen products; that's what is recommended in the Elephant and it looks to
me as close to practice; it's very unlikely to have a nonconstructive proof
of existence of products.

Thomas



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Re: Not invariant but good

[Michael Shulman]

On Sun, Sep 26, 2010 at 10:36 PM, John Baez <baez@math.ucr.edu> wrote:
> Can every property of categories that is invariant under equivalence be
> expressed in some language that doesn't include equations between objects?
> Or conversely? Or what precise conditions are needed to get theorems along
> these lines?

The converse is very easy, and it's something that I and others have
frequently mentioned in these discussions: if we write category theory
in dependent type theory with arrows dependent on their source and
target and no equality predicate on objects, then all formulas and
constructions in this language are easily proven to be invariant under
equivalence and isomorphism.

The forward direction is trickier, but essentially the answer is yes:
I believe theorems along these lines can be found in:

1) Peter Freyd, "Properties invariant within equivalence types of
categories", 1976

2) Georges Blanc, "Équivalence naturelle et formules logiques en
théorie des catégories",  1979

3) Michael Makkai, "First-order logic with dependent sorts, with
applications to category theory,"
http://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf

(and perhaps others that I'm unaware of).

Mike


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Re: Not invariant but good

[John Baez]

Peter wrote:


> In order to focus on the math and not on the terminology, let me today use
> the word "XXXX" instead of "evil".
>

Good.  I think you're secretly on my side, though, because you're using four
X's.

:-)


> I don't think the notion used in your examples is general enough. For
> example, fix some groupoid C, and consider the property of an object
> x: "x is isomorphic to exactly 3 objects of C".


What a fiendishly clever example!


> To me, this is clearly XXXX, because it is not invariant under equivalences
> of
>
C. Yet, according to the definition you used in this email, it extends
> to a functor C -> {F,T}, and therefore is non-XXXX.
>

True.

By the way, in case anyone out there forgets: "Extending to a functor C ->
{F,T}" was merely a pedantic way of saying "being a property of objects of C
that is invariant under isomorphisms" - a pedantic way that lets us easily
generalize this notion to "being a structure on objects of C that is
covariant under isomorphisms".  To generalize, we just replace {F,T} by Set.

It may not be clear to everybody why I like this pedantic approach, so I
should probably explain why.   A (-1)-category is a truth value, a
0-category is a set, and a 1-category is a category.  So, when we replace
{F,T} by Set, we are replacing the 0-category of (-1)-categories by the
1-category of 0-categories.   Since we are just increasing a certain
parameter by 1, it becomes easy to see how to continue this game
indefinitely.

For more details, try this:

http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=12

For a property P of objects x of a category C, "being invariant under
>
  isomorphisms of objects in C" is strictly weaker than "being invariant
> under equivalences of C".


Yes.

Here's my rejoinder:

I had been fixing a groupoid C and asking whether a property of objects of
that category was invariant under isomorphisms.  When you say "x is
isomorphic to exactly 3 objects of C", you are actually treating C not as
fixed but as variable. The more things we let vary, the more invariance
properties we can demand!

In particular, there's a 2-groupoid Cat_* where an object is a "pointed
category" (C,x), that is, a category C with chosen object x.

I can treat "being an object x that is isomorphic to exactly 3 objects in C"
as a property of pointed categories.  And, I would call this property
evil... whoops, I mean XXXX... because it determines a function

Ob(Cat_*)  -> {F,T}

that does not extend to a 2-functor

Cat_* -> {F,T}

Again, this is just a pedantic way of saying what you're saying.  I'm just
trying to point out that I can fit it into my philosophy.


> I tried to give a more general and precise 2-categorical definition on
> the categories list on January 3, 2010, but I am not sure I got it
> quite right.


I remember enjoying that post, but I'll need to reread it to remember what
you said.


> I think it was Mark Weber who also pointed out, around
> the same time, that one person's XXXX concept is another person's
> non-XXXX concept - in a different 2-category.
>

Very much so!  And you've also noticed here that sometimes a property of
objects in a fixed category arises from a property of pointed categories....
so that we can take either a 1-categorical or a 2-categorical approach to
the XXXXness of this property.


> So I don't think it is correct to identify the concept of XXXX with
> "having to talk about equality".
>

I agree.  I take it as a *rule of thumb* that when somebody writes down a
property of categories that involves equations between objects, they're
running the risk that this property is not invariant under equivalence of
categories.   But I don't know the general theorems that make this rule of
thumb precise.

Can every property of categories that is invariant under equivalence be
expressed in some language that doesn't include equations between objects?
Or conversely? Or what precise conditions are needed to get theorems along
these lines?

Best,
jb


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