RE: unsolved problems

RE: unsolved problems

[Hasse Riemann]

 
Hi David

 

Since i could not e-mail you directly, i got delivery failure, i send my response here.

 
>One thing that I have found is that one has to develop one's own feeling for categories.

>I wouldn't say I am terribly good at abstract category theory (monads and algebras and

>Kan extensions and so forth), but I work with categories more in the style of Ehresmann

>- my thesis is essentially on homotopy ideas.
 
I have developed many perspectives on categories
 
As the study of algebraic structures with several objects
As the study of primitive mathematical universe or space (not as fancy as a topos)
As an unifying tool in mathematics
As a foundation of mathematics (that is structural)
As an abstarction of an abstarction of an abstarction of ... (if you go to higher categories)
As a generalized theory of representations 


If i have missed someone please let me know.
 
I don't think i understand in what style Ehresmann worked in.
 
> Here is a real, famous unsolved problem, which Michael Batanin is on his way to solving:
> Prove the homotopy theory of \infty-groupoids is equivalent to the homotopy theory of

> spaces and the related
 
Should it not be weak oo-groupoids?
I think you also mean homotopy category of spaces instead of
"homotopy theory of spaces andthe related".
Spaces is a bit vague but i encounter this sometimes in category theory.
Usually in such statements it is ment a topological space.

Is there a categorical definition of a space (not a topological space)?
 
> But I am sure someone has already mentioned these
> Prove the homotopy theory of n-groupoids is equivalent to the homotopy theory

> of n-types
 
I have seen similar problems and maby this one also.
John Baez mentioned a bunch of such problems on an internet page
and Ronnie Brown also in explaining pursuing stacks.


Thank you for the problems.
 
Best regards
Rafael Borowiecki


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

Re: unsolved problems

[Uwe.Wolter]

Hi Rafael,

> I have developed many perspectives on categories
>
> As the study of algebraic structures with several objects
> As the study of primitive mathematical universe or space (not as   
> fancy as a topos)
> As an unifying tool in mathematics
> As a foundation of mathematics (that is structural)
> As an abstarction of an abstarction of an abstarction of ... (if you  
>  go to higher categories)
> As a generalized theory of representations
>
>
> If i have missed someone please let me know.

I came from Algebraic Specifications to Category Theory. When my  
students (computer science, software engineering) ask me about the  
benefit of categories I'm referring often to the "Categorical  
Manifesto" of Jo Goguen.

If they insist more and ask "Be honest! What is the REAL reason that  
some theoreticians like categories so much?" Then I'm trying to be  
honest and say: Because categories are the winner of the competition  
"What mathematical structure is closed under the maximal number of  
"reasonable" constructions". They have just the right amount of  
structure - not too few, as graphs for example, and not to much, as  
cpo's for example. And if there is time I'm telling them about the  
"Erlanger Programm" of Felix Klein.

Of course we can turn this statement and say: If a construction  
doesn't provide a category when the "inputs" are categories, then this  
construction can not be considered to be a "reasonable construction".

Best regards

Uwe Wolter



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

unsolved problems

[Hasse Riemann]

 

Hello categorists

 

To those who have replied on my question and others.

 

First i want to thank for the suggestions of problems.

 

Since i am new here some people assume that i seek a problem to solve so i can quickly become famous.
I am in fact much more interested in the structure and foundation of mathematics than solving
math problems. I just felt i should know these problems since they are famous, but almost none came to mind.

 

A reason for this is that the proofs i have seen have mostly been "not water tight", they miss some things (mostly in the logic).
And, they are not spelled out in full but are very "cut down" in their arguments. This makes them too time
consuming to follow. I don't want to read a proof to recreate half of it but to understand why something is true.

 

"A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street" -David Hilbert

 

I argue that the same is true for proofs. Something to think about next time you write a proof.

 

In Dyson Freemans terminology i am a bird and not a frog. As a bird i must say that it is sad that about 90% of mathematicians are frogs. This give an inbalance in mathematics.

 

To answer Michael Shulman:

Yes, there is something different about category theory. It is at the top of mathematics. A perfect place for a bird :)

 

Now to my question.

 

After some intensive work and help from people on the mailing list i have found
some problems that are interesting in ordinary category theory :)

 

You just have to learn to see them. Some won't be stated even as problems to solve.

 

To illustrate my point:
How many categories of n objects are there up to categorical equivalence for n a natural number?
If it can not be given directly, can it be expressed by other counting functions?
Maby the pattern is easier than that for groups.

 

A simplified version of the problem is to count only finite categories with at most 1 morphism between any objects.
This suggests the NC(r,s)-problem:
What is the number of categories with r objects and at most s morphisms (in one direction) between any two objects.

 

Specialized versions of these are to count model categories, toposes (defined as categories and not as 2-categories),...
Correct me if it is not so, but i don't know of any theorem that model categories or toposes must be infinite.

 

Another example first stated by John Baez that i call the no-go quantization conjecture:
There is no functor from the symplectic category (symplectic manifolds and symplectomorphisms) to the Hilbert category (Hilbert spaces and unitary operators) that preserves positivity. I.e. a one-parameter group of symplectic transformations generated by a positive Hamiltonian is mapped to a one-parameter group of unitary operators with a positive generator.

 

That maby helps to find the problems. Not that i am trying to popularize them.
If you know some you can still e-mail them to me.

 

Now from the question to the mysterious Hasse Riemann!

 

It seems that people wonder about me so here i go.

This is written from a 15 year old gymnasium account from the time of the beginning of internet.
Bernhard Riemann was my hero then because of riemannian geometry and riemann surfaces.
Actually i have still not managed to replace Riemann!
Hasse is just a name that i think fits me more than Rafael.
Since everyone i e-mail knows me by this pseudo i have decided not to change it.

 

It took me 20 years of studying mathematics to find category theory.
It is a long time but it was hardly wasted time.
I actually went into category theory 3 times before (not so deep), but not until this fourth time
i understood what category theory really is and that it is precisely what i was looking for :)
,and hence decided to stay here for a long while.


The first year in category theory went with a blazing speed.
Now, a half year after that i have more questions than facts.
I happen to be seated in Stockholm. It is not a bad place but there are no category theorists in Sweden!
Hence i'm counting on you people.

 

Best regards

Rafael Borowiecki

 

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