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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeOct 26th 2010
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   Author: Andrew Stacey
   Format: MarkdownItex(Sort of a follow-up to [teaching category theory](http://www.math.ntnu.no/~stacey/CountingOnMyFingers/TeachingCategories.html)) I just taught my students about the minimum polynomial of a linear transformation (in case anyone's interested, the notes are [here](http://www.math.ntnu.no/~stacey/Teaching/TMA4145h2010/lectures.html), lecture 20). When I was thinking how best to explain the whole idea, it struck me that this is an example of how category theory makes it clearer what's really going on. I shan't bore anyone with lots of details, but I thought some might be interested in a real live example of teaching using category theory, so here's the brief outline: 1. Polynomials are examples of "functions without domains" - like certain other "functions", they don't seem to need a particular domain to make sense. 2. Where do they make sense? Anywhere where there's a sensible notion of addition, scaling, and multiplication. 3. In particular, linear transformations of a vector space to itself. 4. Why is this interesting? Knowledge! If we know $p(t)$ and we know $T$ then we know $p(T)$ so we can combine our knowledge of polynomials together with our knowledge of a particular linear transformation to get knowledge of a whole ream of other linear transformations. At which point we go in to trying to figure out how many things we can know about and that leads us to minimum polynomials. The point I'm trying to make is that teaching category theory doesn't have to involve teaching Category Theory. Just because I never said "natural transformation", "category", or "functor" doesn't make those concepts any less prevalent in what I say. Anyway, I thought some here might find that interesting.

   (Sort of a follow-up to teaching category theory)

   I just taught my students about the minimum polynomial of a linear transformation (in case anyone’s interested, the notes are here, lecture 20). When I was thinking how best to explain the whole idea, it struck me that this is an example of how category theory makes it clearer what’s really going on. I shan’t bore anyone with lots of details, but I thought some might be interested in a real live example of teaching using category theory, so here’s the brief outline:

   1. Polynomials are examples of “functions without domains” - like certain other “functions”, they don’t seem to need a particular domain to make sense.
   2. Where do they make sense? Anywhere where there’s a sensible notion of addition, scaling, and multiplication.
   3. In particular, linear transformations of a vector space to itself.
   4. Why is this interesting? Knowledge! If we know $p(t)$ and we know $T$ then we know $p(T)$ so we can combine our knowledge of polynomials together with our knowledge of a particular linear transformation to get knowledge of a whole ream of other linear transformations.

   At which point we go in to trying to figure out how many things we can know about and that leads us to minimum polynomials.

   The point I’m trying to make is that teaching category theory doesn’t have to involve teaching Category Theory. Just because I never said “natural transformation”, “category”, or “functor” doesn’t make those concepts any less prevalent in what I say.

   Anyway, I thought some here might find that interesting.

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeOct 26th 2010
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   Author: Tim_Porter
   Format: MarkdownItexI always found that students seemed to like that sort of approach. It relates to the 'analogy' theme that I have mentioned several time. You appeal to analogy and can even introduce the idea that if it works it is good and it if does not work, but we can see why, that is good as well!

   I always found that students seemed to like that sort of approach. It relates to the ’analogy’ theme that I have mentioned several time. You appeal to analogy and can even introduce the idea that if it works it is good and it if does not work, but we can see why, that is good as well!

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 26th 2010
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   Author: Todd_Trimble
   Format: MarkdownItexI firmly believe that this "sneaky" way of introducing proto-categorical ideas is good pedagogy. When I was teaching, I would routinely do this in, for example, introductory analysis courses where one has to manipulate sets and subsets and logical formulae; I would sneak in adjunctions between posets and the Yoneda lemma (without saying the words) whenever they seemed worthwhile as a general yoga. (In graduate analysis courses at Loyola, I would do more along the lines of universal properties, but again usually without mentioning natural transformations. The better students would realize what I was up to!)

   I firmly believe that this “sneaky” way of introducing proto-categorical ideas is good pedagogy. When I was teaching, I would routinely do this in, for example, introductory analysis courses where one has to manipulate sets and subsets and logical formulae; I would sneak in adjunctions between posets and the Yoneda lemma (without saying the words) whenever they seemed worthwhile as a general yoga. (In graduate analysis courses at Loyola, I would do more along the lines of universal properties, but again usually without mentioning natural transformations. The better students would realize what I was up to!)