Re: Famous unsolved problems in ordinary category theory

Re: Famous unsolved problems in ordinary category theory

[Michael Shulman]

Probably people are going to jump on me for saying this, but it seems to
me that category theory is different from much of mathematics in that
often the difficulty is in the definitions rather than the theorems, and
in the questions rather than the answers.  Thus, there are probably many
unsolved problems in category theory, but we don't know what they are
yet, because figuring out what they are is the main aspect of them
that is unsolved.  (-:

Mike

On Tue, Jun 2, 2009 at 11:31 AM, Hasse Riemann <rafaelb77@hotmail.com> wrote:
>
>
>
> Hello categorists
>
> I don't know what to make of the silence to my question.
> This is the easiest question i have. I can't believe it is so difficult.
> It is not like i am asking you to solve the problems.
>
> There must be some important open problems in ordinary category theory.
> There are plenty of them in the theory of algebras and
> in representation theory, so there should be more of them in category theory.
>
> Especially if you broaden the boundaries a bit of what ordinary category theory is.
> Take for instance:
> model categories,
> categorical logic,
> categorical quantization,
> topos theory-locales-sheaves.
> But i had originally pure category theory in mind.
>
> Best regards
> Rafael Borowiecki
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>


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Re: Famous unsolved problems in ordinary category theory

[Jaap van Oosten]

Reinhard Boerger wrote:
> Dear categorists,
>
> When I read the question for the first time, I did not know such a problem.
> Moreover, my impression was that in category theory one often finds new
> results, which had not been conjectured before. Sometimes an important part
> of the work is even to develop the right notions. This may explain that
> there are less important well-known problems in category theory than in
> other areas.
>
> Nevertheless, I remember a problem that can be easily formulated in pure
> category and is still unsolved as far as I know. Bur it does not seem vastly
> distributed. Cantor's diagonal says that says that the power set always is
> of larger cardinality as the original set. Gavin Wraith suggested the
> following generalization to topoi: If for two objects A,B there is a
> monomorphism A^B>->B, is there also a monomorphism A>->1? This looks like a
> meaningful analogue, and I have not seen an answer in the meantime. The
> question can even be asked not only in a topos, but in every cartesian
> closed category. Does anybody know anything about progress?
>
Dear Professor Boerger,

there are counterexamples to this in elementary topoi. In the effective
topos there are nontrivial objects X such that 2^X is isomorphic to 2
(for example, one can take the object R of real numbers for X; this
gives a mono 2^R>->R), and the inclusion N-->N^{P(N)} is an isomorphism
(giving a mono N^{P(N)}>->P(N) , where P(N) is the power object of N).
I believe the example of 2^R>->R also holds in sheaf toposes where
(internally) the object of functions R^R coincides with the object of
continuous functions (since R is always connected).

Best, Jaap van Oosten
> Greetings
> Reinhard
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>



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Re: Famous unsolved problems in ordinary category theory

[Steve Vickers]

Dear Hasse,

The presheaves that Townsend and I used are on the category Loc of
locales. The fact that Loc is large may be seen as a problem, but
another illegitimacy is the way a presheaf is a functor to Sets. This
means, for instance, that for a representable presheaf y(X), where X is
a locale, we take y(X)(W) = Loc(W,X) to be a _set_ for any pair of
locales X and W, and that is foundationally tendentious. Whatever kind
of collection Loc(W,X) is (if W is locally compact then we can take it
to be another locale, but otherwise not), the ability to extract a "set
of points" from it is sensitive to the foundations.

We tried to be foundationally conservative in what we did with the
presheaves, and you can see come remarks on this in the conclusions of
our paper. (I should stress that we did not claim to have embedded Loc
in a CCC, and we tried not to make use of any particular categorical
properties of Presh(Loc).) Insofar as the representable presheaves y(X)
can be acceptable, then so too are their exponentials y(Y)^y(X), since
y(Y)^y(X)(W) = Loc(WxX,Y). What we showed is that then the exponential
y($)^(y($)^y(X)) also exists (where $ = the Sierpinski locale), and in
fact is representable of the form y(PP(X)) where PP(X) is the "double
powerlocale" on X. Thus PP(X) has a claim to be thought of as $^($^X)
even when X is not exponentiable (locally compact). PP is a
foundationally robust construction, available in both topos-valid locale
theory and predicative formal topology.

Regards,

Steve Vickers.

Hasse Riemann wrote:
>> Another is how to embed the category of locales in a CCC WITHOUT
>> using illegitimate presheaves (Vickers and Townsend) or the axiom
>> of collection (Heckmann).
>
> I don't follow to the end here.
> Why should presheaves be illegitimate?


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Famous unsolved problems in ordinary category theory

[Reinhard Boerger]

Dear categorists,

When I read the question for the first time, I did not know such a problem.
Moreover, my impression was that in category theory one often finds new
results, which had not been conjectured before. Sometimes an important part
of the work is even to develop the right notions. This may explain that
there are less important well-known problems in category theory than in
other areas.

Nevertheless, I remember a problem that can be easily formulated in pure
category and is still unsolved as far as I know. Bur it does not seem vastly
distributed. Cantor's diagonal says that says that the power set always is
of larger cardinality as the original set. Gavin Wraith suggested the
following generalization to topoi: If for two objects A,B there is a
monomorphism A^B>->B, is there also a monomorphism A>->1? This looks like a
meaningful analogue, and I have not seen an answer in the meantime. The
question can even be asked not only in a topos, but in every cartesian
closed category. Does anybody know anything about progress?

Greetings
Reinhard



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Re: Famous unsolved problems in ordinary category theory

[soloviev]

Dear All -

Here a question related to categorical logic (or categorical proof theory)
of a very different type. I would like to put it here because it is an
illustration of another part of the field and also because it is
technically difficult and interesting.

It is well known that certain systems of propositional logic have a
natural structure of free category for certain classes of categories with
structure. For example, we have a structure of free Symmetric Monoidal
Closed Category on the Intuitionistic Multiplicative Linear Logic. In this
structure formulas are objects and equivalence classes of derivations of
the sequents A -> B are morphisms.

Free SMCC (in presence of "tensor unit" I) is not "fully coherent": there
are non-commutative diagrams. For example, one has the Mac Lane's example
A*** -> B***
(called "triple dual diagram"). In terms of IMLL there exist two
non-equivalent derivations of
((A-oI)-oI)-o I -> ((A-oI)-oI)-o I
w.r.t. the equivalence of free SMCC on the derivations of IMLL.  One
derivation is identity, another derivation is obtained in obvious way
using the derivability of ((A-oI)-oI)-o I -> A-o I.  Let us denote these
derivations 1 and f respectively. (For the sequent ((A-oI)-oI)-o I ->
((A-oI)-oI)-o I every derivation is equivalent to 1 or to f.)

The "triple dual conjecture" says that if we declare f\equiv 1 then all
the derivations with the same final sequent in IMLL will become
equivalent. I.e. the stronger categorical structure than SMCC (obtained by
adding this new axiom for equivalence/ commutativity of diagrams) will be
"fully coherent".

If it is true we would have an interesting new variety of categories
(subvariety of SMCCs) in the sense of Universal Algebra.

Proof-theoretically, the study of this conjecture requires to study the
equivalence relations on derivations of IMLL between the relation of free
SMCC and the relation  that  identifies all derivations with the same
final sequent.

In my paper
S. Soloviev. On the conditions of full coherence in closed categories.
Journal of Pure and Applied Algebra, 69:301-329, 1990.

it was shown that
 - if the "triple dual diagram" is commutative w.r.t. some equivalence
relation ~ (containing the relation of free SMCC)
- and the following additional condition holds:
[a-oI/a] h ~ [a-oI/a] g => h~g
for any two derivations of the same sequent,

then all the derivations of the same sequent in IMLL become equivalent.

The additional condition is a) difficult to verify b)  has the form
different form the equational form ("commutativity of a diagram") required
from the point of view of Universal Algebra approach. All the attempts to
prove "pure" triple dual conjecture (by myself and others) did not yet
succeed.

One may mention that it is known that some intermediate equivalence
relations between the relation of free SMCC and the "total" relation of
derivations do exist:
L. Mehats, S. Soloviev. Coherence in SMCCs and equivalences on deriva-
tions in IMLL with unit. Annals of Pure and Applied Logic, v.147, 3, p.
127-179, august 2007.

but all known intermediate relations are contained in the relation
generated by commutativity of triple dual diagram.

My ph.d. student
Antoine El Khoury has checked also that the commutativity of triple
dual diagram (equivalence of 1 and f) implies equivalence of derivations
of the balanced sequents with 1, 2 or 3 variables (commutativity
of corresponding diagrams in SMCC).

Remark. Obviosly, the commutativity of triple dual diagram implies
A*** isomorphe to A*.

Best regards to all

Sergei Soloviev



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Re: Famous unsolved problems in ordinary category theory

[Bhupinder Singh Anand]

On Friday, June 05, 2009 4:37 PM, Ronnie Brown wrote in
categories@mta.ca:

RB>> Mind you there was a serious point: how to turn abstract
mathematics into machine computation? <<RB

Ronnie
======
I shall be presenting the following two papers analysing this problem -
and proposing a solution in the case of the Peano Arithmetic, PA - on
June 14th at the 2009 International Conference on Theoretical and
Mathematical Foundations of Computer Science, Orlando, USA.

TMFCS 173: The significance of Aristotle's particularisation in the
foundations of mathematics, logic and computability II - Goedel and
formally undecidable arithmetical propositions

http://alixcomsi.com/25_Aristotlean_particularisation_II_Goedel_Update.p
df

TMFCS 174: The significance of Aristotle's particularisation in the
foundations of mathematics, logic and computability IV - Turing and a
sound, finitary, interpretation of PA

http://alixcomsi.com/25_Aristotlean_particularisation_IV_Turing_Update.p
df

Regards,

Bhup



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Famous unsolved problems in ordinary category theory

[Hasse Riemann]

 

Hi Paul

 

I still think you are getting me wrong, as did Ronnie. But never mind, i am used to it

since i don't follow the mainstream science ways to specialize, solve problems, publish, repeat.
Yet the problems interest will pass very soon. I now know 19 ordinary category problems (if you explain

this one) vs. at least 23 in higher category theory. This explains why i restricted to ordinary categories. 

 

>From the good side i should be thankful that you and Ronnie trie to direct me towards "true mathematics",

but i have already found my "true mathematics". A big part of the process to get there was precisely to

ask own quastions and finding the answers to them. But some people just got irritated when i asked them

questions (in their field!) they didn't have the answer to.

 

> Another is how to embed the category of locales in a CCC WITHOUT
> using illegitimate presheaves (Vickers and Townsend) or the axiom
> of collection (Heckmann). 

 

I don't follow to the end here.
Why should presheaves be illegitimate?
Then, i suppose the axiom of collection is valid at least in the CCC.
But what is so bad about the axiom of collection in this case?

Do the embedding get bad?

 

Best regards
Rafael Borowiecki



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Re: Famous unsolved problems in ordinary category theory

[Robin Cockett]

A student asked them twice
"Aren't problems just so nice?
They get stuck in your hair
And the last one's just so rare  .."

"Like lice." said they with a grin
"Our hair is all gone and thin ...
And Categories we espouse
Rather than such vermin house!"

-very anon


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Re: Famous unsolved problems in ordinary category theory

[Thomas Streicher]

Dear Rafael,

here is a list of problems from categorical logic that I find difficult
to solve and don't know the answer yet. Certainly they are more on the logical
side but categories are involved in all of them. I don't claim that these
problems are generally important for category theory but they simply do bother
me. I write this mail to show that there are technically hard problems and
with the salient hope that someone may come up with an answer. Nevertheless
I am aware that the subsequent list of problems might easly get a prize for
the "best collection of most misleading problems".

(1) In their booklet "Algebraic Set Theory" Joyal and Moerdijk defined
    for every strongly inaccessible cardinal \kappa a class of
    \kappa-small maps inside the effective topos.

    Does there exist a "generic" \kappa-small map such that all other maps
    can be obtained as pullbacks from this generic one?

    In their book the authors show the existence of a weakly generic one
    but this doesn't imply the existence of a generic one and I suspect
    there is none.

(2) Does there exist a model for Martin-L\"of's Intensional Type Theory
    which validates Church's Thesis?

    This question is due to M.Maietti and G.Sambin. Notice that type theory
    validates the axiom of choice and, accordingly, the statement is much
    stronger than saying that for every function from N to N there exists a
    code for an algorithm computing this function.

(3) In my habilitation thesis
    (www.mathematik.tu-darmstadt.de/~streicher/HabilStreicher.pdf)
    I showed that the sconing of the effective topos (i.e. glueing
    Gamma : Eff -> Set) gives rise to a model of INTENSIONAL Martin-Loef
    type theory faithfully reflecting most of the weakness compared to
    EXTENSIONAL type theory.
    Martin Hofmann and I showed that the groupoid model refutes
    the principle UIP saying that all elements of identity types are equal.
    This has recently generalised to \omega-groupoids by M.Warren and there
    is recently some activity of constructing models based on abstract
    homotopy.

    Can one construct a categorical model model serving both purposes?

    This is an issue since the groupoid model and related ones constructed
    more recently have the defect that all types over N are fairly extensional
    and thus don't do the job which sconing of the effective topos does.

(4) Is the realizability model for the polymorphic lambda calculus
    parametric in the sense of Reynolds?
    It needn't be realizability over natural numbers.
    Would be interesting already for some pca!

Thomas



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Re: Famous unsolved problems in ordinary category theory

[Ronnie Brown]

John,

Glad you liked it! Thanks for the references to Raoul Bott.

Mind you there was a serious point:

how to turn abstract mathematics into machine computation?

I have discussed this often with Larry Lambe.

Ronnie




----- Original Message -----
From: "John Iskra" <jiskra@ehc.edu>
To: "Ronnie Brown" <ronnie.profbrown@btinternet.com>
Cc: "Hasse Riemann" <rafaelb77@hotmail.com>; <categories@mta.ca>
Sent: Friday, June 05, 2009 3:54 AM
Subject: Re: categories: Re: Famous unsolved problems in ordinary category
theory


> One of my favorite quotes:
>
> The question you raise ``how can such a formulation lead to
> computations'' doesn't bother me in the least! Throughout my whole life
> as a mathematician, the possibility of making explicit, elegant
> computations has always come out by itself, as a byproduct of a thorough
> conceptual understanding of what was going on. Thus I never bothered
> about whether what would come out would be suitable for this or that,
> but just tried to understand -- and it always turned out that
> understanding was all that mattered.
>
> A. Grothendieck
>
>
> Raoul Bott reinforced this in a talk I had the privilige to hear back in
> 98.  He said that mathematics, done well, never required the placing of
> your oar in the water (he probably put it better than that...).  The
> idea I think is that if you continually ask and answer the questions
> that occur to you, and, thus, gain understanding, then you will
> inevitably make progress. And that is what matters, really.  So often
> the person credited with solving a 'famous' problem only takes the final
> step in a hard journey of a thousand miles made by a thousand others.
>
> Glory and fame - such as it is in the world of mathematics - are nice,
> but they are not, in the end,  mathematics.  I think it is of high
> importance to avoid confusing them.
>
> John Iskra
>

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Famous unsolved problems in ordinary category theory

[Paul Taylor]

Rafael Borowiecki, under the alias Hasse Riemann, asked,
> Are there any famous unsolved problems in category theory?

Ronnie Brown's posting in response to this is a classic, and
deserves to be printed out and pinned up in every graduate
student's office!   I particularly like the military analogy
with the choice between a frontal assault and making the
obstacle obsolete.   The following point is especially important:

> I was early seduced (see my first two papers) by the idea of
> looking for questions satisfying 3 criteria:
> 1) no-one had previously asked it;
> 2) the question was technically easy to answer;
> 3) the answer was important.
> **** Usually it has been 2) which failed! ****

I sent (a version of) the following reply to "categories" when
Rafael first asked the question, but then asked Bob to withdraw
it as I thought I could write it better.  I put off doing so
because other topics were under discussion,  but by his posting
Ronnie has obliged me to send it, since otherwise I would just
be a chicken.

So here goes:

Do I hear taunts of "do you have a Fields Medal?"?

These are a bit like those of "do you have a girlfriend?".

Well, no, I admit it. I don't.   I have a boyfriend (Richard), and
some of you have met him.   If you bear with me, you will see that
this is not a completely frivolous answer, even though it is a
personal one.

My point is that there are analogies between being a gay man and
being a conceptual--constructive mathematician:

- They both involve long periods of self-doubt and pretence in the
   face of real and perceived discrimation.  This is very much still
   real in the mathematical case, as evidenced by that fact that
   categorists and consructivists are largely to be found in
   computer science departments, excluded from mathematics in case
   they might corrupt the youth.

- The result of this is a significantly delayed adolescence --
   I have met gay men going through adolescence in the 50s or 70s.

- Finally, there is pride in being who you are, and the recognition
   of "Honi soit qui mal y pense" -- that it is the people who think
   ill of it that have the problem.  In the words of a song from
   "La Cage aux Folles" that is known as the "sweet potato song",
   "I yam what I yam!".

Before I came out as a categorist, I pretended to be interested
in difficult puzzles,   I was in the British team in the
International Mathematical Olympiad in 1979, but didn't do very
well.  I started a magazine called QARCH, whose total output
in 30 years amounts to less than one of my papers now.

I was taught as an undergraduate by the Hungarian analyst and
graph theorist Bela Bollobas.  He set problems for first year
students problems that took three weeks to solve, if at all.
(Bela is a mathematician of considerable stature -- so great that
it took me five years to notice that he is 10cm shorter than me --
and I remember him with great affection, in case he gets to read this.)

However, I hope that Bela (along with Andrej Bauer, Imre Leader and
Dorette Pronk, who help organise IMO things in Slovenia, Britain and
Canada nowadays), will forgive me if I say that there is something
fundamentally unsatisfying about IMO problems.   Once you have the
solution, that is it.  They are like crosswords or jigsaws or sudoku.

After that I had my delayed adolescence (with an unsuitable
boyfriend).  I studied continuous posets instead of algebraic
ones and categories instead of posets, just to show that I could.
Somebody should have told me to get a proper job as a programmer,
but they didn't have the guts to say it to me.  (If graduate
students ask me for advice nowadays, I do tell them to get proper
jobs, and not surprisingly they (mis)interpret this personally.)

Long after this, the first paper on Abstract Stone Duality was
published on my 40th birthday, more or less.   According to
G H Hardy's depressing "Mathematician's Apology", and to the rules
for getting a Fields Medal,  I was officially finished as a
mathematician.  But it is pretty clear that I have been doing
my best mathematics during my fifth decade.  On the other hand,
all of those gratuitously difficult problems had gone into the mix.

Before I return to the question.  please refer to number 6 in
     en.wikipedia.org/wiki/Hilbert's_problems
which asks for the axiomatisation of physics.  Even in this most
famous collection of gratuitously difficult problems, we find a
conceptual question.

The first of Hilbert's problems is called the "continuum hypothesis",
but is about smashing the continuum into dust.   Elsewhere, he
said "no-one shall expell us from Cantor's Paradise", but I regard
it as a dystopia.  I dream of some eventual escape, returning to the
Euclidean paradise.  There we would actually talk about lines,
circles, compact subsets or whatever, instead of families of subsets
or arcane algebra (or, indeed, category theory).   I am looking for
a language for mathematics that would look like "set theory" (as
mathematicians, not set theorists, perceive it) but would yields
computable continua instead of dust.

More categorically, I believe that there is some notion of category
that is very similar to an elementary topos, but in which all
morphisms are continuous (in particular Scott continuous with
respect to an intrinsic order).

I also believe that these ideas are applicable to other subjects.
When I have made the appropriate tools, I hope to be able to understand
algebraic geometry, which was a complete mystery to me as a student.
I am in princple capable of doing this, BECAUSE I am a categorist,
by following the analogy between frames and rings.

One version of this problem that I still cannot solve is a question
that Eugenio Moggi asked me in April 1993, although I forget the
exact words.   We wanted a class of monos (I said they should be
the equalisers targetted at power of Sigma) that was closed under
composition and application of the Sigma^2 functor (ie taking the
exponential Sigma^(-) twice).

Another is how to embed the category of locales in a CCC WITHOUT
using illegitimate presheaves (Vickers and Townsend) or the axiom
of collection (Heckmann).  When I wrote the original version of
this posting a couple of weeks back, I thought I could solve this
one.  I am still hopeful, but it turns out to be a powerful question,
cf Ronnie's (2) above.

Notice that I give the principal formulation of the question in
vague language, not as a Diophantine equation.  The more specific
the question, the more likely it is to have been the WRONG one.
Asking an impertinent question is the best way of getting a
pertinent answer.

This still involves very difficult problems and hundreds of journal
pages of formal proofs.  But for me the problems serve the concepts
rather than the other way round.  This is the essence of what it is
to be a conceptual mathematician.   Ronnie Brown has told you a
different story of his own, but with the same message.  Many
other experienced categorists (including the ones in higher
dimensions, which Rafael excluded from his original question,
for some reason) would do likewise.

What about Fields Medals?   People will get them, using my work,
two or three generations down the line.

Paul Taylor





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Famous unsolved problems in ordinary category theory

[John Baez]

Rafael Borowiecki wrote:


> I don't know what to make of the silence to my question.
> This is the easiest question i have. I can't believe it is so difficult.
> It is not like i am asking you to solve the problems.
>
> There must be some important open problems in ordinary category theory.


I think the reason for the silence is that category theory is a bit
different than other branches of mathematics.  Other branches of mathematics
get very excited about patterns that may exist, but may not.  So when
mathematicians hear the phrase "famous unsolved problems", that's the sort
of thing that comes to mind: for example, Goldbach's conjecture, the twin
prime conjecture, the Riemann hypothesis or the Hodge conjecture.

On the other hand, category theorists tend to get excited about taking
already partially understood patterns in mathematics and making them very
clear.   So, the most important open problems often aren't  of the form "Is
this statement true or false?"  Instead, they tend to be a bit more
open-ended, like "Develop a workable theory of n-categories."  So, they
don't have names.

I've tried to encourage people to work on n-categories by emphasizing five
"hypotheses": the homotopy hypothesis, the stabilization hypothesis, the
cobordism hypothesis, the tangle hypothesis, and the generalized tangle
hypothesis.  I didn't want to call them "conjectures", because they're a bit
open-ended.  But they're precise enough that someone can claim to have
proved one, and people can probably agree on whether this has occurred.  For
example, Jacob Lurie claims to have proved the cobordism hypothesis:

http://arxiv.org/abs/0905.0465

http://lab54.ma.utexas.edu:8080/video/lurie.html

and when he provides the full details, people should be able to decide if he
has.

Best,
jb

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Re: Famous unsolved problems in ordinary category theory

[John Iskra]

One of my favorite quotes:

The question you raise ``how can such a formulation lead to
computations'' doesn't bother me in the least! Throughout my whole life
as a mathematician, the possibility of making explicit, elegant
computations has always come out by itself, as a byproduct of a thorough
conceptual understanding of what was going on. Thus I never bothered
about whether what would come out would be suitable for this or that,
but just tried to understand -- and it always turned out that
understanding was all that mattered.

A. Grothendieck


Raoul Bott reinforced this in a talk I had the privilige to hear back in
98.  He said that mathematics, done well, never required the placing of
your oar in the water (he probably put it better than that...).  The
idea I think is that if you continually ask and answer the questions
that occur to you, and, thus, gain understanding, then you will
inevitably make progress. And that is what matters, really.  So often
the person credited with solving a 'famous' problem only takes the final
step in a hard journey of a thousand miles made by a thousand others.

Glory and fame - such as it is in the world of mathematics - are nice,
but they are not, in the end,  mathematics.  I think it is of high
importance to avoid confusing them.

John Iskra

Ronnie Brown wrote:
> In reply to Hasse Riemann's question (see below):
>
> I remember being asked this kind of question at a Topology conference in
> Baku in 1987.  It is worth discussing the background to this, as someone who
> has never gone for a `famous problem', but found myself trying to develop
> some mathematics to express some basic intuitions.
>
> Saul Ulam remarked to me in 1964 at my first international conference
> (Syracuse, Sicily) that a young person may feel the most ambitious thing to
> do is to tackle a famous problem; but this may distract that person from
> developing the mathematics most appropriate to them. It was interesting that
> this remark came from someone as good as Ulam!
>

...

>
>
>
>
>
>
>
> ----- Original Message -----
> From: "Hasse Riemann" <rafaelb77@hotmail.com>
> To: "Category mailing list" <categories@mta.ca>
> Sent: Tuesday, June 02, 2009 5:31 PM
> Subject: categories: Famous unsolved problems in ordinary category theory
>
>
>
>
>
> Hello categorists
>
> I don't know what to make of the silence to my question.
> This is the easiest question i have. I can't believe it is so difficult.
> It is not like i am asking you to solve the problems.
>
> There must be some important open problems in ordinary category theory.
> There are plenty of them in the theory of algebras and
> in representation theory, so there should be more of them in category
> theory.
>
> Especially if you broaden the boundaries a bit of what ordinary category
> theory is.
> Take for instance:
> model categories,
> categorical logic,
> categorical quantization,
> topos theory-locales-sheaves.
> But i had originally pure category theory in mind.
>
> Best regards
> Rafael Borowiecki
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>
>
> --------------------------------------------------------------------------------
>
>
>
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> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
> .
>
>


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Re: Famous unsolved problems in ordinary category theory

[Hasse Riemann]

 

Dear Ronnie
 

This is what stuck with me from the e-mail

 

> One aim of mathematics is understanding, making difficult things easy,
> seeing why something is true. Thus improved exposition is an important part
> of the progress of mathematics (even if this is ignored by Research
> Assessment Exercises). 

 

Indeed they don't teach you this at the university, but somehow i always knew it.

It was obvious from the start, then i had to resist everyone trying to tell me otherwise.

I am glad that there are other who see this as well.

 

> Grothendieck wrote to me in 1982: `The introduction of the cipher 0 or the
> group concept was general nonsense too, and mathematics was more or less
> stagnating for thousands of years because nobody was around to take such
> childish steps ...'.

 

I like this quote since i like structuralizing mathematics.

Something to think about if you want to take the next step.

Fill in the void with a precise mathematical void.

 

Best regards

Rafael Borowiecki

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Re: Famous unsolved problems in ordinary category theory

[tholen]

Finding the "right" questions and notions is certainly a prominent
theme in category theory, perhaps more prominently than in other
fields. Still, just like in other fields, solving open problems was
always part of the agenda. For example, half a century ago people asked
whether every "standard construction" (=monad) is induced by an
adjunction, and it took a few years to have two interesting answers.
And there is ceratinly a string of examples leading all the way to
today.

I don't know whether there are any >famous< unsolved problems in
ordinary category theory, but there are certainly non-trivial
questions. Here is one that we formulated in an article with Reinhard
B"orger (Can. J. Math 42 (1990) 213-229) two decades ago:

A category A is total (Street-Walters) if its Yoneda embedding A --->
Set^{A^{op}} has a left adjoint. Then
1. A has small colimits, and
2. any functor A-->B that preserves all existing colimits of A has a
right adjoint.
Do properties 1 and 2 imply totality for A?

I must admit that, after formulating the question we never considered
it again, so there may well be a known or quick answer. So don't hold
back please, especially since I plan to incorporate several questions
of this type in my CT09 talk.

Walter.



Quoting Michael Shulman <shulman@uchicago.edu>:

> Probably people are going to jump on me for saying this, but it seems to
> me that category theory is different from much of mathematics in that
> often the difficulty is in the definitions rather than the theorems, and
> in the questions rather than the answers.  Thus, there are probably many
> unsolved problems in category theory, but we don't know what they are
> yet, because figuring out what they are is the main aspect of them
> that is unsolved.  (-:
>
> Mike
>

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Re: Famous unsolved problems in ordinary category theory

[Ronnie Brown]

In reply to Hasse Riemann's question (see below):

I remember being asked this kind of question at a Topology conference in
Baku in 1987.  It is worth discussing the background to this, as someone who
has never gone for a `famous problem', but found myself trying to develop
some mathematics to express some basic intuitions.

Saul Ulam remarked to me in 1964 at my first international conference
(Syracuse, Sicily) that a young person may feel the most ambitious thing to
do is to tackle a famous problem; but this may distract that person from
developing the mathematics most appropriate to them. It was interesting that
this remark came from someone as good as Ulam!

G.-C. Rota writes in `Indiscrete thoughts' (1997):
What can you prove with exterior algebra that you cannot prove without it?"
Whenever you hear this question raised about some new piece of mathematics,
be assured that you are likely to be in the presence of something important.
In my time, I have heard it repeated for random variables, Laurent Schwartz'
theory of distributions, ideles and Grothendieck's schemes, to mention only
a few. A proper retort might be: "You are right. There is nothing in
yesterday's mathematics that could not also be proved without it. Exterior
algebra is not meant to prove old facts, it is meant to disclose a new
world. Disclosing new worlds is as worthwhile a mathematical enterprise as
proving old conjectures. "

It is like the old military question:  do you make a frontal attack; or find
a way of rendering the obstacle obsolete?

I was early seduced (see my first two papers) by the idea of looking for
questions satisfying 3 criteria:
1) no-one had previously asked it;
2) the question was technically easy to answer;
3) the answer was important.

Usually it has been 2) which failed!

Of course you do not find such questions where everyone is looking! It could
be interesting to investigate how such questions arise, perhaps by pushing a
point of view as far as it will go, or seeing a new analogy.

"If at first, the idea is not absurd, then there is no hope for it." Albert
Einstein
It could be interesting to investigate historically:

 if (let us suppose) category theory has advanced without a fund of famous
open problems, how then has it advanced?

One aim of mathematics is understanding, making difficult things easy,
seeing why something is true. Thus improved exposition is an important part
of the progress of mathematics (even if this is ignored by Research
Assessment Exercises). R. Bott said to me (1958) that Grothendieck was
prepared to work very hard to make something tautological. By contrast, a
famous algebraic topologist replied to a question of mine about his graduate
text by asking: `Is the function not continuous?' He never gave me a proof!
And I never found it! (Actually the function was not well defined, but that
I could fix!)
Grothendieck wrote to me in 1982: `The introduction of the cipher 0 or the
group concept was general nonsense too, and mathematics was more or less
stagnating for thousands of years because nobody was around to take such
childish steps ...'. See also

http://www.bangor.ac.uk/~mas010/Grothendieck-speculation.html

The point I am trying to make is that the question on `open problems' raises
issues on the nature of,  on professionalism in, and so on the methodology
of, mathematics. It is a good question to start with.

Hope that helps.

Ronnie Brown























----- Original Message -----
From: "Hasse Riemann" <rafaelb77@hotmail.com>
To: "Category mailing list" <categories@mta.ca>
Sent: Tuesday, June 02, 2009 5:31 PM
Subject: categories: Famous unsolved problems in ordinary category theory





Hello categorists

I don't know what to make of the silence to my question.
This is the easiest question i have. I can't believe it is so difficult.
It is not like i am asking you to solve the problems.

There must be some important open problems in ordinary category theory.
There are plenty of them in the theory of algebras and
in representation theory, so there should be more of them in category
theory.

Especially if you broaden the boundaries a bit of what ordinary category
theory is.
Take for instance:
model categories,
categorical logic,
categorical quantization,
topos theory-locales-sheaves.
But i had originally pure category theory in mind.

Best regards
Rafael Borowiecki


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


--------------------------------------------------------------------------------



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17:53:00



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Famous unsolved problems in ordinary category theory

[Hasse Riemann]

 

Hello categorists
 
I don't know what to make of the silence to my question.
This is the easiest question i have. I can't believe it is so difficult.
It is not like i am asking you to solve the problems.
 
There must be some important open problems in ordinary category theory.
There are plenty of them in the theory of algebras and
in representation theory, so there should be more of them in category theory.
 
Especially if you broaden the boundaries a bit of what ordinary category theory is.
Take for instance:
model categories,
categorical logic,
categorical quantization,
topos theory-locales-sheaves.
But i had originally pure category theory in mind.
 
Best regards
Rafael Borowiecki


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

Famous unsolved problems in ordinary category theory

[Hasse Riemann]

Hi all categorists
 
Here are other questions i think about and need your help with.
 
4>
Are there any famous unsolved problems in category theory not related to higher dimensional category theory
(but monoidal categories are ok as categories)?
 
Best regards
Rafael Borowiecki