Re: Category Theory for the Sciences

Re: Category Theory for the Sciences

[Fred E.J. Linton]

Older, and shorter. The print version runs to over 400 pp. whereas that web
PDF stays under 300.

Still the web version faithfully conveys the flavor of the print version.  Just
see how late, and how lackadaisically, Yoneda's Lemma enters into the fray.

Cheers, -- Fred 

---

------ Original Message ------
Received: Fri, 30 Jan 2015 08:56:40 AM EST
From: "Garraway, Dale" <dgarraway@ewu.edu>
To: Michael Barr <barr@math.mcgill.ca>,        
  Categories mailing list	<categories@mta.ca>
Subject: categories: Re: Category Theory for the Sciences

> 
> 
> There is apparently a slightly older version online.  It can be found at
> 
>
http://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013/textbook/MIT18_S996S13_textbook.pdf
> 
> 
> 
> Dale Garraway
> Dept. of Math,
> Eastern Washington
> dgarraway@ewu.edu
> 
> 
> 
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Re: Category Theory for the Sciences

[Vaughan Pratt]

On 1/30/2015 11:03 AM, Fred E.J. Linton wrote:
> Just see how late, and how lackadaisically, Yoneda's Lemma enters into [Spivak's book].

As those who heard my CT'2011 talk (which grew out of my CT'2004 talk on
communes) may recall, I'm in favour of exploiting the Yoneda Lemma in
the way automobile manufacturers exploit the internal combustion engine:
not as something whose mechanism is to be understood but merely as a
means of propulsion controlled by the accelerator pedal.

Such an approach could potentially make it accessible to more than just
physicists, in particular to a wide range of workers in the social sciences.

To that end, define a Sigma-category (C,Sigma) to be any category C
equipped with a distinguished set Sigma of objects of C.  In the obvious
(to this audience) way, this determines a multisorted unary theory T,
namely T = J' as the opposite of the full subcategory J of C with ob(J)
= Sigma.

In T, the objects represent the sorts and the morphisms the operations
of the theory.

In C, every object represents some model of T and every morphism
represents some homomorphism of those models, not necessarily faithfully
(a homomorphism may have more than one representative).

What I find particularly appealing about this presentation of
multisorted unary theories and (some of) their models and homomorphisms
is that it extends so straightforwardly to Sigma-Pi-categories
(C,Sigma,Pi).  Here Pi is a second subset of ob(C) dual to Sigma in the
sense that

(a) whereas Sigma consists of the *sorts* of T, Pi consists of its
*properties*; and

(b) whereas the *elements* of a model M are the morphisms from Sigma to
M, with a: s --> M being an element of sort s, the *states* of M are the
morphisms from M to Pi, with x: M --> p being a state for property p.


[Two asides:

1.  There is a nice alliteration pun here between the duality of sorts
and properties and that between sums and products.

2.  Up to equivalence there is an obvious notion of maximal
Sigma-Pi-category subject to leaving elements and states invariant.
With that notion, the following special cases arise:

(i) for Pi empty: the presheaf category Set^T;

(ii) for Sigma = {I}, Pi = {_|_}, rigid in the sense that |C(I,I)| =
|C(_|_,_|_)| = 1: the (ordinary) Chu category Chu(Set, C(I, _|_)); and

(iii) for Sigma = Pi: what Bill Lawvere has called the Isbell envelope
E(J) (J as above).]


One social science that could find Sigma-Pi-categories useful is
philosophy, which could find in them a single mathematical home for all
three of the problems of

(a) Cartesian dualism;

(b) extensionality of properties (as the sets of states of a model); and

(c) logical consistency of qualia (as morphisms from Sigma to Pi).

(Slightly) more on this application in Section 3.3 of Fundamenta
Informaticae 103 (2010) 203?218 at

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.395.2995&rep=rep1&type=pdf

Vaughan Pratt


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Re: Category Theory for the Sciences

[Fred E.J. Linton]

Dale Garraway <dgarraway@ewu.edu> noticed that, in re Category Theory for  the
Sciences,

> There is apparently a slightly older version online.  It can be found at
> 
>
http://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013/textbook/MIT18_S996S13_textbook.pdf
... .

That's dated Sept. 17, 2013. MIT hosts earlier versions as well, viz.:

http://math.mit.edu/~dspivak/teaching/sp13/CT4S--static.pdf of Feb. 5, 2013

and

http://math.mit.edu/~dspivak/teaching/sp13/CT4S.pdf of May 14, 2013,

all related to a course of his entitled "Category Theory for Scientists"
(the printed book is entitled, instead, "Category Theory for the Sciences").

Cheers, -- Fred




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Category Theory for the Sciences

[Michael Barr]

A book of that name by David I. Spivak, Mathematics at MIT was recently 
published by the MIT Press.  Has anyone seen it?  Did it seem interesting. 
I wonder what kind of science outside of string theory would find CT 
useful.

Michael


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Re: Category Theory for the Sciences

[Marco Benini]

Dear Michael,
I've received it a couple of days ago. I should write a review for
Zentralblatt math.

I've not yet seen how it applies CT to other sciences - apparently,
string theory does not appear in the book.
It is an introduction to (basic) CT with many examples and exercises
which try to apply the fundamental concepts in other fields.
As far as I've seen, there are many applications to computer science,
with some emphasis on databases. But, I had just a quick glimpse by
now...

All the best,
M


Dr Marco Benini
Università degli Studi dell'Insubria
marco.benini@uninsubria.it
http://marcobenini.wordpress.com/
http://marcobeniniphoto.wordpress.com/


On 29 January 2015 at 01:59, Michael Barr <barr@math.mcgill.ca> wrote:
> A book of that name by David I. Spivak, Mathematics at MIT was recently
> published by the MIT Press.  Has anyone seen it?  Did it seem interesting.
> I wonder what kind of science outside of string theory would find CT
> useful.
>
> Michael
>
>


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Re: Category Theory for the Sciences

[Harley Eades III]

Hi, Michael.

I thought I would mention one more thing.  I think that this book and other work
is just the beginning of a whole new area of applied CT.  Perhaps I am wrong.

However, there are a few groups of people working on new applications using
primarily CT.

For example, check out the recent work of John Baez:

http://math.ucr.edu/home/baez/networks/index.html

John and his students have been making progress studying network theory
and control theory using CT.  This includes electrical circuits and chemical 
reactions.

Also, check out the recent report 

"Report from Dagstuhl Perspectives Workshop 14182: Categorical Methods at the Crossroads”

http://vesta.informatik.rwth-aachen.de/opus/volltexte/2014/4618/pdf/dagrep_v004_i004_p049_s14182.pdf

This was a meeting to discuss using CT as the basis for math modeling and 
applied science.

Very best,
Harley

On Jan 29, 2015, at 8:41 AM, Harley Eades III <harley.eades@gmail.com> wrote:

> Hi, Michael.
> 
> On Jan 28, 2015, at 7:59 PM, Michael Barr <barr@math.mcgill.ca> wrote:
> 
>> A book of that name by David I. Spivak, Mathematics at MIT was recently 
>> published by the MIT Press.  Has anyone seen it?  Did it seem interesting. 
>> I wonder what kind of science outside of string theory would find CT 
>> useful.
> 
> I have been reading it a little here and there.  I find it interesting, and fun
> to read.
> 
> You can find an older draft of the book on the authors webpage:
> 
> http://math.mit.edu/~dspivak/CT4S.pdf
> 
> If you are curious.
> 
> The books is an introduction to CT, but with an eye towards applications in
> the sciences.
> 
> It is based, I think, on the intuition the author has obtained from his
> work on using category theory to study databases.  He uses these ideas to
> come up with a nice illustrative way to relate categorical — and other mathematical — 
> ideas to various scientific situations called ontology logs (ologs).  These are essentially
> database schemes or a diagrams in CT.  However, they are more informal.  Then given an
> olog we can talk about facts, which are just commutative diagrams.   You can see a bunch
> of examples in the book.  These ologs help take an application one has in mind and situate it
> so the categorical structure is illuminated.
> 
> I find it interesting.  I really like his chapter on spans where he uses them to model
> experiments and metrics. 
> 
> As for sciences he talks about computer science, information science, chemistry, physics,
> material sciences.  I can’t recall which others.  
> 
> Very best,
> Harley
> 
>> 
>> Michael
>> 


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Re: Category Theory for the Sciences

[Charles Wells]

*Higher* category theory has been of great interest to a certain kind of
physicist, not just string theorists, for some years. The central place
where they commune is at  the n-category café
<https://golem.ph.utexas.edu/category/> and its ancillary site n-lab
<http://ncatlab.org/nlab/show/HomePage>. Lately some higher category
theorists have done a lot of work on homotopy type theory.

Charles

On Wed, Jan 28, 2015 at 6:59 PM, Michael Barr <barr@math.mcgill.ca> wrote:

> A book of that name by David I. Spivak, Mathematics at MIT was recently
> published by the MIT Press.  Has anyone seen it?  Did it seem interesting.
> I wonder what kind of science outside of string theory would find CT
> useful.
>
> Michael
>

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Re: Category Theory for the Sciences

[Garraway, Dale]

There is apparently a slightly older version online.  It can be found at

http://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013/textbook/MIT18_S996S13_textbook.pdf



Dale Garraway
Dept. of Math,
Eastern Washington
dgarraway@ewu.edu




On 2015-01-28, 4:59 PM, "Michael Barr" <barr@math.mcgill.ca<mailto:barr@math.mcgill.ca>> wrote:

A book of that name by David I. Spivak, Mathematics at MIT was recently
published by the MIT Press.  Has anyone seen it?  Did it seem interesting.
I wonder what kind of science outside of string theory would find CT
useful.

Michael


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Re: Category Theory for the Sciences

[majordomo]

Michael,

Cognitive Neuroscience, if we’re successful in surviving dissociative experiments which we don’t have funding for at present (lots of cutbacks and competition for funds).  Here are some refs; if interested, I can send:

This one has a link:
M. J. Healy and T. P. Caudell (2006a) 
Ontologies and Worlds in Category Theory: Implications for Neural Systems, 
Axiomathes, vol. 16, nos. 1-2, pp. 165-214.

M. J. Healy, R. D. Olinger, R. J. Young, S. E. Taylor, T. P. Caudell, and K. W. Larson (2009) 
Applying Category Theory to Improve the Performance of a Neural Architecture, 
Neurocomputing, vol. 72, pp. 3158-3173. 

Anothr with a link:
M. J. Healy, T. P. Caudell, and T. E. Goldsmith (2008) 
A Model of Human Categorization and Similarity Based Upon Category Theory,
UNM Technical Report EECE-TR-08-0010, DSpaceUNM, University of New Mexico. 

There’s more, including a beginning at addressing episodic memory.  We’ve done quite a bit on that more recently and haven’t had time to write it all up.  Also, Tom and I are on ResearchGate.

Best regards,
Mike

On Jan 28, 2015, at 5:59 PM, Michael Barr <barr@math.mcgill.ca> wrote:

> A book of that name by David I. Spivak, Mathematics at MIT was recently 
> published by the MIT Press.  Has anyone seen it?  Did it seem interesting. 
> I wonder what kind of science outside of string theory would find CT 
> useful.
> 
> Michael
> 

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Re: Category Theory for the Sciences

[Ronnie Brown]

On 29/01/2015 00:59, Michael Barr wrote:
> I wonder what kind of science outside of string theory would find CT
> useful.
I think there are lots of answers to Mike's question. Just to give one,
I gave a talk in 2003 published with Tim Porter as

   `Category theory and higher dimensional algebra: potential descriptive
tools in neuroscience', Proceedings of the International Conference on
Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh,
National Brain Research
Centre, Conference Proceedings 1 (2003) 80-92. arXiv:math/0306223

which went well. A web search on "category theory and biology" shows
lots more.

Category theory has (at least) two aspects.  One as a kind of meta
theory for discussing and relating mathematical structures; another as
giving a range of algebraic,  or more generally,  mathematical,
structures, often with algebraic operations with partial domains. We can
expect these to have over time surprising applications

Ronnie

-- 



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Re: Category Theory for the Sciences

[Patrik Eklund]

On 2015-01-29 02:59, Michael Barr wrote:
> I wonder what kind of science outside of string theory would find CT
> useful.

Dear Michael,

Years ago I was in touch with you on monad compositions and monads over
something else than just Set. Uncertainty modelling has been interesting
for us, and monads over monoidal cats are important, because then we can
generalize the signature in a useful way.

See e.g.

http://www.sciencedirect.com/science/article/pii/S0165011413000997

Let me mention "health" and health nomenclatures as an area, not
restricting it only to science, where CT is powerful. Health ontology
has been "infected" by simplistic things like description logic, which
is just a relational view, so CTwise its just the powerset monad over
Set. It's awful to see how SNOMED thinks "ontology" in "health ontology"
is the same as "ontology" in "web ontology". However, when we really
start to investigate the structure e.g. of WHO's (World Health
Organization) reference and derived classifications, we find term monad
based approaches very useful. Work is still in its infancy, but as
Shakespeare's number of lives is seven, we have six to go, and we are
approaching childhood, we think.

Those of the readers who know a bit of these classifications already
know what I am talking about, and for those who don't, let me just
mention a simple example on the distinction between "co-morbidity" and
"multimorbidity". Setwise speaking it's a set of ICD codes, but since we
do not want to drop that "co", we have an (pre)order between those
codes. Further, it's a hierarchy, so it requires a "powertype", and I am
not convinced HoTT treats these things properly. We believe it requires
a "level of signatures" not tried out before.

If anyone is interested, I can organize a short virtual presentation
over Adobe Connect to explain this "application area".

Best,

Patrik


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Re: Category Theory for the Sciences

[Andree Ehresmann]

Dear Michael,

In relation with "Category Theory for the Sciences", I would mention
the amplications of category theory I have been developing since more
than 25 years to the modeling of multi-scale living systems, such as
biological, cognitive and social systems.
Cf. my book with Jean-Paul Vanbremeersch
"Memory Evolutive Systems: Hierarchy, Emergence, Cognition", Elsevier 2007
and for more recent papers my site
http://ehres.pagesperso-orange.fr

This work is partially based on an extension of sketch theory, leading
to a characteization of the properties necessary for 'real' emergence.
Among the applications: model MENS for a neuro-cognitive-mental
system, from the neural level to mental processes, analyzing the
emergence of higher cognitive processes up to consciousness,
anticipation and creativity.

Kind regards
Andree






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