Category Theory for the Sciences
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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_v... 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
*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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
There is apparently a slightly older version online. It can be found at http://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientist... 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 -- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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-scientist...
Dale Garraway Dept. of Math, Eastern Washington dgarraway@ewu.edu
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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-scientist... ... . 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (11)
-
Andree Ehresmann -
Charles Wells -
Fred E.J. Linton -
Garraway, Dale -
Harley Eades III -
majordomo＠mlist.mta.ca -
Marco Benini -
Michael Barr -
Patrik Eklund -
Ronnie Brown -
Vaughan Pratt