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 =)
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.
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.