# Category Theory.

## Main Question or Discussion Point

Essentially I'd just like to learn more about it in my own spare time. Is there any particularly good introductory books or articles?

I'd particularly like anything that uses examples from physics, but it isn't essential.

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There's alway http://en.wikipedia.org/wiki/Category_theory" [Broken]. Dummitt and Foote has a decent intro to category theory as an appendix.

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John Baez is one of the experts in category theory who often posts to his blogs 'n-category Cafe' or 'This Week’s Finds in Mathematical Physics'.

Week 245 has talks in this theory from a U-Toronto workshop at the Fields Institute January 9-13, 2007.
http://math.ucr.edu/home/baez/week245.html

matt grime
Homework Helper
John Baez is not an expert in category theory.

Mac Lane is the standard book, but it is very hard going.

mathwonk
Homework Helper
i liked peter freyd's little book in the 60's and 70's.

hungerford ahs a short little section that helps too.

but the amin point is to make yourself always focus on the maps betwen objects instead of just the obejects.

e.g. from a pair of spaces one can form a product space. but also from maps of two pairs of spaces, obe derives maps of their products.

]this si thew whole point. which constructions of spaces allow comparable constructions of maps between those spaces?

the fundamental group assigns to a space maps from a circle into that space. but also think abiout how a map of the spaces indiuces a map of their fundamental groups. now you are thinking "categorically".

mathwonk
Homework Helper
the fact that the product of determinants is the determinsnt of the product matrix says that a determinant isa functor from amtrices to numbers.

the chain rule says that the derivative is a fucntor from pointed functions to numbers.

at some point people realized that amny important constructions were fucntors. but to=defione functors, betwen categories, one ahd to define categories.

i recomend the original apper by maclane and who?

matt, what is the original paper defining exact sequences and so on, and especially natural transformations, with the double dual as the basic example? was it by eilenberg and maclane?

is this it?

General Theory of Natural Equivalences
S Eilenberg, S MacLane - Transactions of the American Mathematical Society, 1945 - JSTOR
General Theory of Natural Equivalences. Samuel Eilenberg. Saunders MacLane.
Transactions of the American Mathematical Society, Vol. 58, No. 2, 231-294. ...

it seems to be available free on the internet.

do you think this is a good one matt?

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I learnt some category theory from a book called "The Joy Of Cats" by Adamek, Herrlich and Strecker. Its a fairly decent book. But then i also had a good instructor to go along, so i am not sure, how good it might be for self study.

-- AI

mathwonk
Homework Helper
categories are nothing. functors are something. natural transformstions are more interesting still.

So I guess you really love categories of functors with natural transformations as the morphisms.

Incidentally, in a homological algebra course, it was after talking about this and bi-, tri, ... , infinity-categories, that my professor remarked about people calling category theory abstract nonsense.

matt grime
Homework Helper
Even us category theory advocates use that term. It just means: true for elementary formal reasons. Often things are proven in some concrete case in a very inobvious way, but are true for for other simpler reasons in some manner. I.e. in the more general case the proof is actually easier.

John Baez is not an expert in category theory.

Mac Lane is the standard book, but it is very hard going.
I got out Mac Lane on your suggestion. I must say this is an incredibly interesting branch of mathematics. I think it's a really "clean" way to view concepts from other areas.

the fact that the product of determinants is the determinant of the product matrix says that a determinant is a functor from matrices to numbers.
Thanks for that example. Solidified the concept a bit for me.

I was just wondering, in what other areas of mathematics has category theory been useful. (I'd assume it has been useful in Algebraic Topology.)

matt grime
Homework Helper
All parts of algebra (so that is geometry and topology too). Some parts of analysis. Theoretical physics. Computer Science. Is that enough? Few parts of mathematics are not touched by category theory.

Silly examples:

a group is a category with one object and all morphisms invertible. A representation of a group is a functor from this category to the category of vector spaces. An isomoprhism of representations is a natural transformation of functors.

completion of a normed vector space is left adjoint to the forgetful functor from banach spaces to normed vector spaces.

classifications of things are functors (moduli spaces). There is a functor from the genus 2 curves to abelian varieties of dimension 2 (an equivalence, right, roy? curve to jacobian, jacobian to the theta divisor or some other such slogan).

The representation theory of a field is essentially the same as the representation theory of nxn matrices over that field - Morita equivalence.

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All parts of algebra (so that is geometry and topology too). Some parts of analysis. Theoretical physics. Computer Science. Is that enough?
Yeah. Computer Science surprised me to be honest. Of course though, it doesn't matter how "useful" it is. I was just idly wondering how much it had seeped into the general language of mathematicians.

a group is a category with one object and all morphisms invertible. A representation of a group is a functor from this category to the category of vector spaces. An isomoprhism of representations is a natural transformation of functors.

completion of a normed vector space is left adjoint to the forgetful functor from banach spaces to normed vector spaces.

classifications of things are functors (moduli spaces). There is a functor from the genus 2 curves to abelian varieties of dimension 2 (an equivalence, right, roy? curve to jacobian, jacobian to the theta divisor or some other such slogan).

The representation theory of a field is essentially the same as the representation theory of nxn matrices over that field - Morita equivalence.
Thanks for the examples. I really like the representation theory example.

Hi matt grime

Could you clarify your comment in your post of 02-13-2007, 06:41 PM?

I am confused because why would so many university math / physics workshops employ a non-expert to discuss category theory.

Counterparts of yours in ‘Beyond the Standard Model’ [Physics] and others who comment in that forum appear to regard the individual as an expert in category theory as it relates to physics.

Hurkyl
Staff Emeritus
Gold Member
I was just idly wondering how much it had seeped into the general language of mathematicians.
You know how you denote the domain and range of a function with the notation $f : D \rightarrow R$? (AFAIK) Before category theory, people wrote $f \subseteq D \times R$.

matt grime
Homework Helper
I am confused because why would so many university math / physics workshops employ a non-expert to discuss category theory.
knowing about some aspect of something doesn't make you an expert in the subject as a whole

Counterparts of yours in ‘Beyond the Standard Model’ [Physics] and others who comment in that forum appear to regard the individual as an expert in category theory as it relates to physics.

John is very knowledgeable about n-categories and how they pertain to physics. Most mathematicians are under your rules therefore experts in categories since they use them 'expertly' in their own field. That does not in any reasonable sense as far as I am concerned make them experts in category theory.

I know how string theory relates to algebra. That does not make me an expert in physics, or even string theory.

Gib Z
Homework Helper
Not to mention, a member posting in the Physics section is more likely to be biased to the physicist.

Hurkyl: My Teacher still uses that. He either even more ancient than I thought, or just old fashioned.

Introduction to Category theory

Essentially I'd just like to learn more about it in my own spare time. Is there any particularly good introductory books or articles?

I'd particularly like anything that uses examples from physics, but it isn't essential.
I have written a website introducing category theory using examples in physcis.

Hope you find it useful: http://topos-physics.org" [Broken]

If you have any questions use the comments on the site or message me

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matt grime

1 - Thank you for the clear and concise explanation of expertise in your post of 02-16-2007, 04:55 AM.

I agree that it is necessary and likely sufficient for expertise to be qualified as specifically as possible.

2 - I have become very interested in Game Theory because of what I perceive to be an extremely powerful analytic tool, particularly as used by Basar [engineer] and Olsder [mathematician].

Elements of Game Theory are used “... general search algorithm for movement strategies based on the detection of sporadic cues and partial information ...” in the Editor's Summary, 25 January 2007 of Nature 25 January 2007 Volume 445 Number 7126, pp339:
Letter: 'Infotaxis' as a strategy for searching without gradients
Massimo Vergassola, Emmanuel Villermaux and Boris I. Shraiman
doi:10.1038/nature05464
http://www.nature.com/nature/journal/v445/n7126/edsumm/e070125-10.html

I am somewhat surprised that the only mathematics ever to win a Nobel Prize is not listed in within the category of Mathematics. Even though the Nobel category was Economics, only a minor tweak should be needed for use in Energy Economics. The ‘Set Theory, Logic, Probability, Statistics’ appears to be an appropriate forum.

I have posted this thread on this forum:
Game Theory - applied mathematics - how powerful is it?