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Owen Densmore

Category theory - Wikipedia, the free encyclopedia

Administrator

We've knocked around the term Category Theory a bit lately, so I
started looking into it a bit. This seems to be a reasonable
starting place:
http://en.wikipedia.org/wiki/Category_theory

Has anyone used this in complexity science work? Or semantic web
work? Or anything else? :)

I know Amazon turns up Russell Standish's book first in a search for
category theory!

-- Owen

Owen Densmore http://backspaces.net
"You can do Anything, but not Everything!"

Russell Standish

Category theory - Wikipedia, the free encyclopedia

On Sat, Feb 10, 2007 at 02:48:08PM -0700, Owen Densmore wrote:

What?

Some say law is an ass. Does this mean Amazon search is an ass? I
mention category theory only briefly as a more general theory than set
theory, but my work is entirely based on set theory. It would not be
much help to anyone wanting to understand CT.

Cheers

--

----------------------------------------------------------------------------
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpcoder at hpcoders.com.au
Australia http://www.hpcoders.com.au
----------------------------------------------------------------------------

Marcus G. Daniels

Category theory - Wikipedia, the free encyclopedia

In reply to this post by Owen Densmore

Owen Densmore wrote:
> We've knocked around the term Category Theory a bit lately, so I
> started looking into it a bit.
With Language Integrated Query (LINQ), Microsoft has brought monads,
closures, and type inference to the upcoming Visual Studio (Orcas).
They seem to have a good chunk of the functional programming brain trust
at MSR, and now its diffusing to their products..

http://blogs.msdn.com/charlie/archive/2007/01/26/anders-hejlsberg-on-linq-and-functional-programming.aspx
http://msdn2.microsoft.com/en-us/netframework/aa904594.aspx
http://www.microsoft.com/downloads/details.aspx?familyid=1ff0b35d-0c4a-40b4-915a-5331e11c39e6&displaylang=en

http://en.wikipedia.org/wiki/Monad_(category_theory)
http://en.wikipedia.org/wiki/Monads_in_functional_programming

Mikhail Gorelkin

Category theory - Wikipedia, the free encyclopedia

In reply to this post by Owen Densmore

Here is a gentle (conceptual) introduction into the category theory
------------------------------------------------------------------

http://www.cs.toronto.edu/~sme/presentations/cat101.pdf
http://en.wikipedia.org/wiki/Natural_transformation
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
http://www.informatics.bangor.ac.uk/public/math/research/ftp/cathom/05_10.pdf

there is something in "Life Itself" by Robert Rosen

Basic Category Theory for Computer Scientists (Foundations of Computing) by
Benjamin C. Pierce

Categories for Software Engineering by Jos? L. Fiadeiro

Category Theory for Computing Science by Michael Barr and Charles Wells
(it's not gentle at all :-))

There are several articles of J. Goguen about connection between category
theory and general systems theory:

1) J. Goguen (1973) Categorial foundations for general systems theory
2) J. Goguen, S. Ginali (1978) A categorial approach to general systems
theory
3) J. Goguen (1991) A categorial manifesto

+
http://users.viawest.net/~keirsey/princomplexica.html (here is a connection
between complexica and the category theory)
http://www.ralph-abraham.org/articles/MS%23108.Complex/complex.pdf

--Mikhail

----- Original Message -----
From: "Owen Densmore" <[hidden email]>
To: "The Friday Morning Applied Complexity Coffee Group" <friam at redfish.com>
Sent: Saturday, February 10, 2007 4:48 PM
Subject: [FRIAM] Category theory - Wikipedia, the free encyclopedia

> We've knocked around the term Category Theory a bit lately, so I
> started looking into it a bit. This seems to be a reasonable
> starting place:
> http://en.wikipedia.org/wiki/Category_theory
>
> Has anyone used this in complexity science work? Or semantic web
> work? Or anything else? :)
>
> I know Amazon turns up Russell Standish's book first in a search for
> category theory!
>
> -- Owen
>
> Owen Densmore http://backspaces.net
> "You can do Anything, but not Everything!"
>
>
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
>

Carl Tollander

Category theory - Wikipedia, the free encyclopedia

I strongly second the Mazur paper - it's going to important to get a
good handle on
what Category Theory folks mean by equivalence.

Carl

Mikhail Gorelkin wrote:

> Here is a gentle (conceptual) introduction into the category theory
> ------------------------------------------------------------------
>
> http://www.cs.toronto.edu/~sme/presentations/cat101.pdf
> http://en.wikipedia.org/wiki/Natural_transformation
> http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
> http://www.informatics.bangor.ac.uk/public/math/research/ftp/cathom/05_10.pdf
>
> there is something in "Life Itself" by Robert Rosen
>
> Basic Category Theory for Computer Scientists (Foundations of Computing) by
> Benjamin C. Pierce
>
> Categories for Software Engineering by Jos? L. Fiadeiro
>
> Category Theory for Computing Science by Michael Barr and Charles Wells
> (it's not gentle at all :-))
>
> There are several articles of J. Goguen about connection between category
> theory and general systems theory:
>
> 1) J. Goguen (1973) Categorial foundations for general systems theory
> 2) J. Goguen, S. Ginali (1978) A categorial approach to general systems
> theory
> 3) J. Goguen (1991) A categorial manifesto
>
> +
> http://users.viawest.net/~keirsey/princomplexica.html (here is a connection
> between complexica and the category theory)
> http://www.ralph-abraham.org/articles/MS%23108.Complex/complex.pdf
>
> --Mikhail
>
> ----- Original Message -----
> From: "Owen Densmore" <owen at backspaces.net>
> To: "The Friday Morning Applied Complexity Coffee Group" <friam at redfish.com>
> Sent: Saturday, February 10, 2007 4:48 PM
> Subject: [FRIAM] Category theory - Wikipedia, the free encyclopedia
>
>
>
>> We've knocked around the term Category Theory a bit lately, so I
>> started looking into it a bit. This seems to be a reasonable
>> starting place:
>> http://en.wikipedia.org/wiki/Category_theory
>>
>> Has anyone used this in complexity science work? Or semantic web
>> work? Or anything else? :)
>>
>> I know Amazon turns up Russell Standish's book first in a search for
>> category theory!
>>
>> -- Owen
>>
>> Owen Densmore http://backspaces.net
>> "You can do Anything, but not Everything!"
>>
>>
>>
>>
>> ============================================================
>> FRIAM Applied Complexity Group listserv
>> Meets Fridays 9a-11:30 at cafe at St. John's College
>> lectures, archives, unsubscribe, maps at http://www.friam.org
>>
>>
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
>
>
>
>

Carl Tollander

Category theory - Wikipedia, the free encyclopedia

In reply to this post by Owen Densmore

Suggest taking a look at Gougen
http://www-cse.ucsd.edu/users/goguen/ps/manif.ps.gz
(see also Mikhail's references).

or any of the earlier Baez stuff. I particularly like:
http://math.ucr.edu/home/baez/planck/node5.html as a quick introduction.

Stanford: http://plato.stanford.edu/entries/category-theory/

Learning Lounge on Monads: http://tunes.org/wiki/Monads_101

Most of the Cat Theory action I see is in mathematical physics and
computer science. You can start to see some
semantic web stuff show up in http://math.ucr.edu/home/baez/qg-spring2004/
(particularly week 1, though they don't call it that).

FP folks are moving things in interesting ways, but the FP concerns seem
to me to be
a relatively small part of the activity overall. (Don't get me wrong,
type inferencing is
cool).

IMHO, n-Cats, (see mostly Baez, but also
http://golem.ph.utexas.edu/category/)
is where I think the use of CT in complexity applications will show up,
initially
for model composition and recombination.

Carl

Owen Densmore wrote:

Mikhail Gorelkin

Category theory - Wikipedia, the free encyclopedia

In reply to this post by Carl Tollander

know only one software company which uses categories - ATX Software,
Portugal. Interesting articles are here:
http://www.atxsoftware.com/?sec=article

P.S. A good book about math: "Meta Math! The Quest for Omega" by Cregory
Chaitin (a mathematician from IBM Research and a friend of Stephen Wolfram)

--Mikhail

----- Original Message -----
From: "Carl Tollander" <[hidden email]>
To: "The Friday Morning Applied Complexity Coffee Group" <friam at redfish.com>
Sent: Saturday, February 10, 2007 7:33 PM
Subject: Re: [FRIAM] Category theory - Wikipedia, the free encyclopedia

I strongly second the Mazur paper - it's going to important to get a
good handle on
what Category Theory folks mean by equivalence.

Carl

Mikhail Gorelkin wrote:

> Here is a gentle (conceptual) introduction into the category theory
> ------------------------------------------------------------------
>
> http://www.cs.toronto.edu/~sme/presentations/cat101.pdf
> http://en.wikipedia.org/wiki/Natural_transformation
> http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
> http://www.informatics.bangor.ac.uk/public/math/research/ftp/cathom/05_10.pdf
>
> there is something in "Life Itself" by Robert Rosen
>
> Basic Category Theory for Computer Scientists (Foundations of Computing)
> by
> Benjamin C. Pierce
>
> Categories for Software Engineering by Jos? L. Fiadeiro
>
> Category Theory for Computing Science by Michael Barr and Charles Wells
> (it's not gentle at all :-))
>
> There are several articles of J. Goguen about connection between category
> theory and general systems theory:
>
> 1) J. Goguen (1973) Categorial foundations for general systems theory
> 2) J. Goguen, S. Ginali (1978) A categorial approach to general systems
> theory
> 3) J. Goguen (1991) A categorial manifesto
>
> +
> http://users.viawest.net/~keirsey/princomplexica.html (here is a
> connection
> between complexica and the category theory)
> http://www.ralph-abraham.org/articles/MS%23108.Complex/complex.pdf
>
> --Mikhail
>
> ----- Original Message -----
> From: "Owen Densmore" <owen at backspaces.net>
> To: "The Friday Morning Applied Complexity Coffee Group"
> <friam at redfish.com>
> Sent: Saturday, February 10, 2007 4:48 PM
> Subject: [FRIAM] Category theory - Wikipedia, the free encyclopedia
>
>
>
>> We've knocked around the term Category Theory a bit lately, so I
>> started looking into it a bit. This seems to be a reasonable
>> starting place:
>> http://en.wikipedia.org/wiki/Category_theory
>>
>> Has anyone used this in complexity science work? Or semantic web
>> work? Or anything else? :)
>>
>> I know Amazon turns up Russell Standish's book first in a search for
>> category theory!
>>
>> -- Owen
>>
>> Owen Densmore http://backspaces.net
>> "You can do Anything, but not Everything!"
>>
>>
>>
>>
>> ============================================================
>> FRIAM Applied Complexity Group listserv
>> Meets Fridays 9a-11:30 at cafe at St. John's College
>> lectures, archives, unsubscribe, maps at http://www.friam.org
>>
>>
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
>
>
>
>

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Robert Howard-2-3

Category theory - Wikipedia, the free encyclopedia

In reply to this post by Owen Densmore

This is the book I have that an Introduction of Category Theory. Probably
the best book I've read on the subject. Cambridge Press-not difficult-lots
of examples and pictures!

http://www.amazon.co.uk/Conceptual-Mathematics-First-Introduction-Categories
/dp/0521478170

SHORT SUMMARY: Instead of defining a set by the elements it has (the
objects), define the rules for the elements (the properties) and deduce the
elements of the set. This prevents Russell's Paradox.

Robert Howard
Phoenix, Arizona

-----Original Message-----
From: [hidden email] [mailto:[hidden email]] On Behalf
Of Owen Densmore
Sent: Saturday, February 10, 2007 2:48 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: [FRIAM] Category theory - Wikipedia, the free encyclopedia

We've knocked around the term Category Theory a bit lately, so I

started looking into it a bit. This seems to be a reasonable

starting place:

http://en.wikipedia.org/wiki/Category_theory

Has anyone used this in complexity science work? Or semantic web

work? Or anything else? :)

I know Amazon turns up Russell Standish's book first in a search for

category theory!

-- Owen

Owen Densmore http://backspaces.net

"You can do Anything, but not Everything!"

============================================================

FRIAM Applied Complexity Group listserv

Meets Fridays 9a-11:30 at cafe at St. John's College

lectures, archives, unsubscribe, maps at http://www.friam.org

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Steve Smith

Category theory - Wikipedia, the free encyclopedia

In reply to this post by Owen Densmore

Owen, et alii -

>
> Has anyone used this in complexity science work? Or semantic web
> work? Or anything else? :)

My colleagues at UNM... Tom Caudell (whom I believe you have met), Tim
Goldsmith (Cognitive Psychologist) and Mike Healey (Mathematician
retired from UWash) are using it to develop knowledge models from
expert elicitation. The methodology they are developing is essentially
(apologies to them for any mistakes I make) as follows:

1) Collect a set of potential "experts".
2) Interview them about the topic in question, primarily asking what
words (terms) they use ot describe the topic, think about the topic,
pontificate on the topic.
3) Pile all these terms on a big blanket out in the field on a windy
day.
4) Toss the terms in the air and let the wind carry away the
lightweight and trivial ones.
5) Sort through the remainders and join up synonyms .
6) Go back to the experts and ask them to rank the pairwise distance
between terms. (N squared!) One gets a fully connected graph.
7) Do some kind of normalization thingy amongst the results... call it
a numerical average for now.
8) Threshold the edges such that the graph no longer is fully connected
(black magic mojo).
9) Iteratively consult a subset ( the more cooperative ones?) of
experts on steps 7, 8.
10) Viola!

Although I am only peripherally involved in their discussions on this,
I believe:
A) 8) There are probably more advanced graph theoretic things to do
than simply threshold the weights... like collapsing cycles and/or
finding some heirarchy, and/or thresholding some more interesting???
derived measure than the simple, original weights... maybe...
B) 7) There are likely somewhat interesting things to do here,
especially to (later) place the different experts "point of view"
relative to the collective. There would seem to be a lot of soft
and/or unknown factors regarding the nature of the experts... etc.

I'm trying to converge my own less formal theories about Metaphor in
Information Visualization (formal analogy, etc) with their work, but
there is still a bit of distance (probably entirely in my lack of
understanding of the nuances of category theory). My now-30-year old
BS in Mathematics and Physics with a handful of graduate courses in
group theory and topology tossed on top for garnish serves me just well
enough to get in trouble...

I have been doing work in Visualization of Ontologies which also seems
to relate... I'm not sure anyone knows how to build an ontology
really... or how to describe the caveats and conditions surrounding the
Ontology. The Gene Ontology I have worked most with seems to have
plenty of anomolies of both history and of the compromises made to
bring it to a single, agreed-upon ontology...

It seems that most Ontologies, at least for the moment are going to be
self-organizing... that the only people both able and willing to build
such a huge abstract beast are those who will also use it...

One problem (in my opinion) is that it is somewhat of a "theory of
everything" so in some sense, all formal knowledge models can be
expressed in or traced back to category theory... so merely saying that
one is "using category theory" is not unlike replying to the question
"How did you get here?" with "I used a mode of transportation".

For example, at a meeting between Caudell and two of my more strongly
mathematically inclined colleagues last week, it was stated with
complete confidence and agreement around the the table that Formal
Concept Analysis is "just a specific use of Category Theory"...

> We've knocked around the term Category Theory a bit lately, so I
> started looking into it a bit. This seems to be a reasonable
> starting place:
> http://en.wikipedia.org/wiki/Category_theory

Wikipedia strikes again! I am constantly amazed at how accessible and
thorough technical information on Wikipedia is. I can't vouch for it's
accuracy (or thoroughness) in this case, but I am impressed at how well
these articles seem to summarize what I think I already know and plenty
I'm still trying to figure out.

And to make it even more interesting... isn't Wikipedia a
self-organizing ontology of everything? If one "labels" the links
used in Wikipedia to other Wikipedia elements with the verbs used in
the text, does that not begin to make an ontology?

Like the first line In Categories:

mathematics, categories allow one to formalize notions involving
abstract structure and processes that preserve structure.

We have a link between "Categories" and "Mathematics" and perhaps
(suggesting new links or topic are needed in Wikipedia) "Notions" and
"Structure" or perhaps "Abstract Structure" and "Processes", etc..

I look forward to the evolution of this discussion here (If I can even
keep up).

- Steve
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Günther Greindl

Category theory - Wikipedia, the free encyclopedia

Hello,

I keep being surprised by this list, when topics which I have been
interested in for some time come cropping up in the discussion here :-)

I would like to second Robert's recommendation for the book Conceptual
Mathematics, a good place to go from there would be:

Category Theory
http://www.oup.com/uk/catalogue/?ci=9780198568612

from Steve Awodey:
http://www.andrew.cmu.edu/user/awodey/

I haven't read the latter yet (I'm still working throught Conceptual
Math.)but this is definitely the book I will go for next - Awodey has a
good reputation for being accessible without making compromises
regarding content (the book is rather expensive unfortunately).

The classic on Category Theory is

Mac Lane: Categories for the Working Mathematician.

but, as the title suggests, is for fully-fledged mathematicians only ;-)

Regards,
G?nther

steve smith wrote:

> Owen, et alii -
>
>>
>> Has anyone used this in complexity science work? Or semantic web
>> work? Or anything else? :)
>
>
> My colleagues at UNM... Tom Caudell (whom I believe you have met), Tim
> Goldsmith (Cognitive Psychologist) and Mike Healey (Mathematician
> retired from UWash) are using it to develop knowledge models from expert
> elicitation. The methodology they are developing is essentially
> (apologies to them for any mistakes I make) as follows:
>
> 1) Collect a set of potential "experts".
> 2) Interview them about the topic in question, primarily asking what
> words (terms) they use ot describe the topic, think about the topic,
> pontificate on the topic.
> 3) Pile all these terms on a big blanket out in the field on a windy day.
> 4) Toss the terms in the air and let the wind carry away the lightweight
> and trivial ones.
> 5) Sort through the remainders and join up synonyms .
> 6) Go back to the experts and ask them to rank the pairwise distance
> between terms. (N squared!) One gets a fully connected graph.
> 7) Do some kind of normalization thingy amongst the results... call it a
> numerical average for now.
> 8) Threshold the edges such that the graph no longer is fully connected
> (black magic mojo).
> 9) Iteratively consult a subset ( the more cooperative ones?) of experts
> on steps 7, 8.
> 10) Viola!
>
> Although I am only peripherally involved in their discussions on this, I
> believe:
> A) 8) There are probably more advanced graph theoretic things to do than
> simply threshold the weights... like collapsing cycles and/or finding
> some heirarchy, and/or thresholding some more interesting??? derived
> measure than the simple, original weights... maybe...
> B) 7) There are likely somewhat interesting things to do here,
> especially to (later) place the different experts "point of view"
> relative to the collective. There would seem to be a lot of soft
> and/or unknown factors regarding the nature of the experts... etc.
>
> I'm trying to converge my own less formal theories about Metaphor in
> Information Visualization (formal analogy, etc) with their work, but
> there is still a bit of distance (probably entirely in my lack of
> understanding of the nuances of category theory). My now-30-year old
> BS in Mathematics and Physics with a handful of graduate courses in
> group theory and topology tossed on top for garnish serves me just well
> enough to get in trouble...
>
> I have been doing work in Visualization of Ontologies which also seems
> to relate... I'm not sure anyone knows how to build an ontology
> really... or how to describe the caveats and conditions surrounding the
> Ontology. The Gene Ontology I have worked most with seems to have
> plenty of anomolies of both history and of the compromises made to bring
> it to a single, agreed-upon ontology...
>
> It seems that most Ontologies, at least for the moment are going to be
> self-organizing... that the only people both able and willing to build
> such a huge abstract beast are those who will also use it...
>
> One problem (in my opinion) is that it is somewhat of a "theory of
> everything" so in some sense, all formal knowledge models can be
> expressed in or traced back to category theory... so merely saying that
> one is "using category theory" is not unlike replying to the question
> "How did you get here?" with "I used a mode of transportation".
>
> For example, at a meeting between Caudell and two of my more strongly
> mathematically inclined colleagues last week, it was stated with
> complete confidence and agreement around the the table that Formal
> Concept Analysis is "just a specific use of Category Theory"...
>
>
>> We've knocked around the term Category Theory a bit lately, so I
>> started looking into it a bit. This seems to be a reasonable
>> starting place:
>> http://en.wikipedia.org/wiki/Category_theory
>
> Wikipedia strikes again! I am constantly amazed at how accessible and
> thorough technical information on Wikipedia is. I can't vouch for it's
> accuracy (or thoroughness) in this case, but I am impressed at how well
> these articles seem to summarize what I think I already know and plenty
> I'm still trying to figure out.
>
> And to make it even more interesting... isn't Wikipedia a
> self-organizing ontology of everything? If one "labels" the links used
> in Wikipedia to other Wikipedia elements with the verbs used in the
> text, does that not begin to make an ontology?
>
> Like the first line In Categories:
>
> mathematics, categories allow one to formalize notions involving
> abstract structure and processes that preserve structure.
>
> We have a link between "Categories" and "Mathematics" and perhaps
> (suggesting new links or topic are needed in Wikipedia) "Notions" and
> "Structure" or perhaps "Abstract Structure" and "Processes", etc..
>
> I look forward to the evolution of this discussion here (If I can even
> keep up).
>
> - Steve
>
>
> ------------------------------------------------------------------------
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org