I am a mathematician with background in Category Theory. I have been asked to give a 20 minute talk about my area of research to an audience of talented high school students and school mathematicians (not researchers). The audience will have seen notions of abstract mathematics like groups, rings and topological spaces but mathematical maturity can definitely not be assumed. Any ideas on what I could be talking about, in the general area of category theory, that might be interesting to them? I do not want to give a boring talk describing rigorously what a category is, what a functor is, etc... Any help will be much appreciated; I am really struggling =)

  • $\begingroup$ Maybe talk about the applications and motivations of category theory.. perhaps something about its relation to computer science and programming $\endgroup$ – spin Jan 26 '14 at 17:40
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    $\begingroup$ I'm not sure this is a useful suggestion (I like Michael Hardy's) but when I was in secondary school, I was deeply inspired when I was shown the categorial construction of product objects, and the way that the product of two sets could be characterized without any reference to ordered pairs or even to elements. I think you could explain this in twenty minutes, and that it might go a long way to giving the audience the flavor of category theory: the way it refers only to external properties of the objects, for example, and the way it characterizes products, but only up to unique isomorphism. $\endgroup$ – MJD Jan 26 '14 at 17:48
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    $\begingroup$ I would draw a lot of known things as commutative diagrams, thereby emphasizing that "arrows" are more important then "elements". $\endgroup$ – Martin Brandenburg Jan 26 '14 at 17:49
  • $\begingroup$ I agree with what Martin said. The first examples that come to mind are lcm/gcd of integers and sup/inf of real numbers. There are many other elementary examples like these. $\endgroup$ – Bruno Stonek Jan 26 '14 at 17:58

Google "Schanuel" and "category theory". You'll find a book he wrote, Conceptual Mathematics: A First Introduction to Categories, that seems to be an attempt to introduce secondary-school students to category theory.

  • $\begingroup$ is it "Samuel" Eilenberg, right? $\endgroup$ – janmarqz Jan 26 '14 at 17:51
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    $\begingroup$ @janmarqz : No. Eilenberg was one of the founders of category theory. "Shanuel" is the surname of the author of the book I have in mind. $\endgroup$ – Michael Hardy Jan 26 '14 at 17:52
  • $\begingroup$ oh it is [ en.wikipedia.org/wiki/Stephen_Schanuel ] then $\endgroup$ – janmarqz Jan 26 '14 at 17:54
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    $\begingroup$ I believe that the book Michael Hardy is thinking of is Conceptual Mathematics: A First Introduction to Categories by Lawvere and Schanuel. $\endgroup$ – MJD Jan 26 '14 at 17:58

Once you know that your audience has been exposed to certain basic theories like groups and topological spaces in a axiomatic-deductive style and without trying to be too technical you could say that a Category is a wider fashion to organize maths and relate them.

It is like a continuation of the ideas when one finds in the general guiding methods in set theory: there are set and there are maps to relate them. In category is almost the same: there are objects and there are functors to relate them.


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