Open problems

Open problems

[Harley Eades III]

Hi, everyone.

Dee Roytenberg’s email pushed me to write an email I have been wanting to write
for a bit.

One thing I have been trying to do recently is figure out what the major (and minor)
open problems are with respect to applications of category theory to CS.   I am very
new to this area, and getting an idea of what folks are working on, and what problems
people feel are important will help young researchers learns where to concentrate 
their efforts.

I am thinking perhaps we could compile a list of research topics/open problems in
this area.  I hope some of the experts who read this mailing list would like to add 
their thoughts.

To start us off here are a few that I know of:

1. Homotopy type theory
     - The study of a new interpretation of intensional type theory based in homotopy 
        theory. This line of research has a lot of interesting problems that need to be
        solved, and as applications in the design and analysis of functional programming 
        languages, and software verification. 

2. Databases
     - David Spivak has done a lot of great work applying category theory to the design
       and analysis of databases.  I am not sure what the current major open problems are.

3. Security
     - Dusko Pavlovic currently has a number of projects related to security that use category
       theory either explicitly or are inspired by categorical concepts.  I have read a few of
       his papers in this line of work, but I am not sure of the current open problems.  A list
       of the papers in this line were posted here awhile ago and can be found here: 
          http://article.gmane.org/gmane.science.mathematics.categories/8068/match=

What other topics or specific open problems do others know about?  

I look forward to hearing about more.

Very best regards,
Harley Eades

 




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

Re: Open problems

[Vaughan Pratt]

On 12/12/2014 1:56 PM, Harley Eades III wrote:
> Dee Roytenberg?s email pushed me to write an email I have been wanting to write
> for a bit.
>
> One thing I have been trying to do recently is figure out what the major (and minor)
> open problems are with respect to applications of category theory to CS.   I am very
> new to this area, and getting an idea of what folks are working on, and what problems
> people feel are important will help young researchers learns where to concentrate
> their efforts.

At http://thue.stanford.edu/puzzle.html can be seen a problem about Chu
spaces that's been open for close to two decades, and that's been
translated during the past three years into Belorussian, Ukrainian,
Polish, Slovenian, Czech, French, and Bulgarian by volunteers who
apparently found the problem sufficiently interesting to be worth the
effort, perhaps motivated by its connection to crossword puzzles.

The problem is whether every T1 comonoid in Chu(Set,2) is discrete.

This is so for two special cases.

1. Countable comonoids, taking cardinality of a Chu space to be that of
its points.

2. Representable comonoids (of any cardinality), in the sense of
representation by directed CPOs as per the hierarchy of full embeddings
on page 6 of the paper "Comonoids in chu: a large cartesian closed
sibling of topological spaces" at
http://boole.stanford.edu/pub/comonoids.pdf .  T1 DCPOs are of course
discrete simply because T1 posets are discrete and the structure of any
DCPO is determined by its order via the Scott topology (as distinct from
the Alexandroff topology, though they coincide in the T1 case).

The problem remains open for uncountable comonoids not representable as
DCPOs.

The provenance of Chu spaces is arguably category-theoretic, the
enriched case Chu(V,k) having been invented by Mike Barr and first
studied in detail in the late 1970s by his masters student Po-Hsiang
(Peter) Chu.   As a computer scientist my own application for the
ordinary or set-enriched case Chu(Set,K) has been to concurrency theory,
starting around 1991 shortly after Lafont and Streicher proposed that
case (under the rubric of "games") as a very nice model of Girard's
linear logic.

In the latter application the image of Girard's exponential construct !A
is naturally modeled as the cartesian closed full subcategory of that of
Chu spaces consisting of comonoids.  In any symmetric monoidal category
with tensor product A@B and unit I, a comonoid is a structure with a
comultiplication
d: A --> A@A
and a counit
e: A --> I.
A comonoid homomorphism is any morphism preserving that structure.

In Set, taking tensor @ to be cartesian product x and I to be 1 makes
every set a comonoid by taking d to be the diagonal and e the unique map
to 1, and those are the only comonoids in Set.  Every function is
thereby a comonoid homomorphism.

The standard tensor @ of Chu(V,k) is given by

A@B ~ (A --o B')' ~ (B --o A')'           (1)

where A' is the dual or transpose of A and --o is the external hom of
Chu(V,k) internalized by suitably "chupologizing" it.  The unit is that
of V likewise chupologized.

For Chu(Set,K), a Chu space A = (A,X) can be understood as a space A
whose points are positions for letters, and (the points of) its dual A'
as a set (so these are actually *extensional* Chu spaces) X of words of
length A over the alphabet K, with each word formed by filling each
position with a letter from K, i.e. functions A --> K, making X a subset
of K^A.  Duality then requires that A' also be extensional ("little Chu"
or chu(Set,K) in Barr's terminology).

  From that perspective (A@B)' can be visualized as the space of all AxB
crosswords that can be formed by taking the across words from B' (given
by a map from A to B') and the down words from A' (given by a map from B
to A'), as per the two isomorphisms in (1) above, the second of which
amounts to an adjointness condition on the two maps.

The comonoids in Chu(Set,K) are those Chu spaces A for which every
crossword in (A@A)' has for its main diagonal a word in A', and such
that A' includes all one-letter (constant) words aaa...aa, bbb...bb, ....

The T1 comonoids can be defined as those comonoids A such that every
word of length A begun by writing down two of its letters (at any two
positions) can be completed to a word in A'.  For any alphabet K, the
unit makes this automatic when the two letters are the same, so for the
alphabet {0,1} T1 reduces to: at any two given positions both
...0...1... and ...1...0... can be completed to a word.  With the
interpretation of words over {0,1} as open sets this is recognizable as
one definition of a T1 topological space (the other topological
definition of T1, every singleton closed, does not give the chupological
definition.)

So, is every T1 comonoid in Chu(Set,2) discrete?

Vaughan Pratt


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

Re: Open problems

[Michael Barr]

I just wanted to add to what Vaughan said about the provenance of Chu 
spaces.  When I started working on duality in 1975 at the ETH, one 
prominent example was topologicalvector spaces.  So I looked at several 
books on the subject, one by Grothendieck.  They all mentioned a 
construction of pairs (V,V') of spaces equipped with a bilinear map V x V' 
--> K (the ground field).  This simple example, with its obvious duality 
led pretty directly to the Chu construction.  It was amazing that it 
turned to not only give and obvious duality but also a natural 
*-autonomous structure (that Peter Chu studied).

At some point I decided to track down the source.  It certainly had an air 
of Grothendieck about it, but no, it turned out to go back to George 
Mackey's thesis, published in TAMS around 1944, I think.  He looked at 
pairs (V,V') as above and assumed they were extensional but not 
necessarily separated.  He was aware of the duality in the separated 
extensional case, but didn't pursue it in any detail.

So what was the motivation.  I can guess that he was thinking that V', as 
a space of functionals on V, coud replace the topology.  Just as in a 
locally compact abelian group, you could recover the topology if you knew 
which characters were "admissible".  I once wrote him to ask if that was 
his original motivation, but he died within the ensuing year, not having 
answered me.

But truly, it goes back to Mackey.

Michael

----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: "Categories mailing list" <categories@mta.ca>
Sent: Sunday, 14 December, 2014 1:12:58 PM
Subject: categories: Re: Open problems

On 12/12/2014 1:56 PM, Harley Eades III wrote:
> Dee Roytenberg?s email pushed me to write an email I have been wanting to write
> for a bit.
>
> One thing I have been trying to do recently is figure out what the major (and minor)
> open problems are with respect to applications of category theory to CS.   I am very
> new to this area, and getting an idea of what folks are working on, and what problems
> people feel are important will help young researchers learns where to concentrate
> their efforts.

At http://thue.stanford.edu/puzzle.html can be seen a problem about Chu
spaces that's been open for close to two decades, and that's been
translated during the past three years into Belorussian, Ukrainian,
Polish, Slovenian, Czech, French, and Bulgarian by volunteers who
apparently found the problem sufficiently interesting to be worth the
effort, perhaps motivated by its connection to crossword puzzles.

The problem is whether every T1 comonoid in Chu(Set,2) is discrete.

This is so for two special cases.

1. Countable comonoids, taking cardinality of a Chu space to be that of
its points.

2. Representable comonoids (of any cardinality), in the sense of
representation by directed CPOs as per the hierarchy of full embeddings
on page 6 of the paper "Comonoids in chu: a large cartesian closed
sibling of topological spaces" at
http://boole.stanford.edu/pub/comonoids.pdf .  T1 DCPOs are of course
discrete simply because T1 posets are discrete and the structure of any
DCPO is determined by its order via the Scott topology (as distinct from
the Alexandroff topology, though they coincide in the T1 case).

The problem remains open for uncountable comonoids not representable as
DCPOs.

...


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