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Category Theory for Dummies [pdf]
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- antimora 15y agoI am curious how many people know about the Category Theory? I asked one of the Math professors and he had no idea, though we was teaching Group Theory at the time.
- zitterbewegung 15y agoWhich college? Note that category theory is heavily used in topology but homological algebra is used in abstract algebra (group theory). See http://en.wikipedia.org/wiki/Category_theory http://en.wikipedia.org/wiki/Category_theory
- DanielRibeiro 15y agoIt is also used in eliptic and hyperelliptic curve cryptography, when you study the Jacobian Variety a curve: http://modular.math.washington.edu/edu/Fall2003/252/lectures/10-15-03/10-15-03-lecture.pdf http://modular.math.washington.edu/edu/Fall2003/252/lectures... Categories and Functors come out naturally when studying such structures.
- eldina 15y agoI think category theory is rarely essentially used in a first or second typical course on some topic in math. Once in a while if students can be assumed to have had a course in category theory, then it is more elegant and efficient e.g. simply to show that some functor has an adjoint hence this or that limit is preserved, but to me category theory, except if it is the main topic of interest, is useful because it eases communication and allows you to quickly get some understanding of a construction that might look very "local" to some category that you are not familiar with. Unlike the person who created the cited notes, I think it actually can help understanding the things under study or the associated constructions, assuming sufficient command of category theory. I remember when I first understood the definitions of things like products, coproducts, push-outs and pull- backs etc. in my last year as an undergraduate. Suddenly, for many of the constructions from topology and algebra it became easier to remember them, how they were constructed and which properties they had. To me it is kind of like with design patterns, mainly I don't use them as tools picked up from my tool box when solving a problem, rather they simply allow me to communicate more easily and gives me another level of abstraction where I can reuse thinking I have done earlier.
- eldina 15y agoNot being a native speaker, I have to ask: Is a "professor" not someone who at least majored in math ?
- groovy2shoes 15y agoA professor is a "high-ranking" teacher at a university. You can have a professor in pretty much any field, so there are math professors.
- eldina 15y agoOk. Then I think it is a little sad that a math professor can now nothing about category theory.
- xyzzyz 15y agoWhy? Would you say the same thing about, say, ergodic theory, languages, automata and computation theory, analytic number theory, harmonic analysis, bordism and cobordism, algebraic K-theory, de Rham cohomology, Diophantine approximation, model theory and theory of forcing, descriptive set theory, algebraic geometry, representation theory... I could go on like this forever. People tend not to know that most of the stuff a math graduate learned was already known by 30s, and category theory was only starting to become anything close to popular in 50s - 60s. People also tend not to know just how much math research have been done, especially in last 70 years.
- gtani 15y agoslide 2 And Category Theory is not even taught here.
- petrilli 15y agoThis is a heretofore unknown definition of "dummies".
- charlieflowers 15y agoDid you know that mathematicians refer to Category Theory as "General Abstract Nonsense"? It's true. And they mean it as a compliment! It is entirely possible that Category Theory is a practical joke that the mathematicians are playing on the rest of us.
- MaysonL 15y agohttp://en.wikipedia.org/wiki/Abstract_nonsense http://en.wikipedia.org/wiki/Abstract_nonsense
- antimora 15y agoHere is a great "An introduction to Category Theory for Software Engineers" http://www.cs.toronto.edu/~sme/presentations/cat101.pdf http://www.cs.toronto.edu/~sme/presentations/cat101.pdf
- T_S_ 15y agoThanks for that link. It is much better than the one headlined.
- antimora 15y agoOut of curiosity, I did some search on Category Theory popularity on Google's "books ngrams" and found it peaked around 1990 and started picking up again in 2000 Here is the source: http://ngrams.googlelabs.com/graph?content=category+theory&year_start=1800&year_end=2011&corpus=0&smoothing=3 http://ngrams.googlelabs.com/graph?content=category+theory&#...
- DanielRibeiro 15y agoFor a not so short, and way scarier introduction: Physics, Topology, Logic and Computation: A Rosetta Stone[1] However it doesn't shy away from the fact that the sets of all sets is not set, and therefore we need a higher level of abstraction to talk about these non-set things. The same, of course, applies to other set like structures: All sets do not form a set, but form a category All monoids do not form a set, but form a category All graphs do not form a set, but form a category [1] http://math.ucr.edu/home/baez/rosetta.pdf http://math.ucr.edu/home/baez/rosetta.pdf
- antimora 15y agoThank you. It seems like a good resource - very interesting.
- xyzzyz 15y agoIt seems that category theory is the new hype -- almost nobody actually understands what it is about, or, more importantly, what it is for. Let me tell you what it is for -- it's an important technical tool in mathematical research, which gives you new, coherent language, sometimes provides you with an additional insight in the structure of the stuff you are researching and makes it easier to notice and classify similarities between different kind of structures. Unfortunately, it is almost completely useless and uninteresting by itself -- because, well, what's interesting in objects and arrows anyway? What make category theory interesting are its connections with various field and math and computer science. That's why introducing "category theory for dummies" makes completely no sense -- it's like following Erlangen program to teach kids about points, lines and circles on a plane. The need and the significance of Erlangen program arise when you learn about many different geometries, notice what they have in common and what they do not, and try to find out what the geometry is all about. Without it, the Erlangen program is all about abstract bullshit, and the situation is completely the same with category theory. But nobody writes or posts Erlangen program for dummies on HN. Why? "General theory of everything" hype, that's why. Erlangen program is "general theory of geometry", but geometry seems a bit pale when compared to everything. If you really want to understand the significance of category theory, then learn set theory, then algebra, then topology, then algebraic topology and algebraic geometry, or take abstract programming languages theory path. If you don't care about all this stuff, because you're hyped on the category theory, then you're missing the point -- it's like you wanted to learn about algebraic topology, but did not care about algebra or topology. Also anything that has "for dummies" in title should invariably remind you of Norvig's essay (google Peter Norvig 21 days). Also my old comments about category theory on HN: http://news.ycombinator.com/item?id=2713315 http://news.ycombinator.com/item?id=2713315 http://news.ycombinator.com/item?id=2713510 http://news.ycombinator.com/item?id=2713510
- more_original 15y agoCategory theory more than just a language for classifying and organizing existing results. It is the mathematical theory of universal properties (http://en.wikipedia.org/wiki/Universal_property http://en.wikipedia.org/wiki/Universal_property). In category theory one studies what structure can be captured by universal properties. That is one tries to characterizes structure uniquely by its properties, rather than giving a concrete construction, as is common in set theory. The emphasis on universal properties leads to a different way of thinking and mathematical taste. The study of what one can do in general with universal constructions is thus interesing in itself, though perhaps not for dummies. Anyway, what I want to say is that while category theory has been overhyped in the context of Haskell, for example, it is not true that it is "completely useless and uninteresting by itself".