Category Theory is like Set Theory, but supposedly better.  What is it, though?

  • It’s a collection of points, and arrows.
  • Unlike in Set Theory, things can’t just be there, without a meaning attached.
  • No wonder they call it “abstract nonsense"…

An example, then.  Maybe you noticed this when you were learning arithmetic:

I’ll write o for odd and e for even.

o+o=e \\ e+e=e \\ o+e=o \\ e+o=o
(If that confuses you: grab a pen & paper and make substitutions using the identity o = e+1 until you’re satisfied.)

I’ll write Pos for positive and Neg for negative.

N×N=P \\ P×P=P \\ N×P=N \\ P×N=N


Maybe you see it already.  Negative numbers play the role in multiplication that odd numbers play in addition.  Similarly, positive numbers serve the same function in multiplication that even numbers serve in addition.

Namely, × positive and + even preserve the state of the thing they operate on, and × negative and + odd change the state.

Interchanging (e,o,+) for (P,N,×) is an example of a functor.  Sounds like function, but it maps categories to categories.

(e,o,+) for (P,N,×)

One more thing: notice that both of these are isomorphic to the cyclic group Z₂, with even or positive as the identity element.

Just to review. In plain English:  “What evens and odds do in addition, positives and negatives do in multiplication.”  In Category Theory:  “There is an isomorphic functor between the categories {even, odd, +}and {positive, negative, ×}.”


So what’s the big deal?  There is a philosophical difference between Sets and Categories:  Categories require that the relationships between the objects come along for the ride.  I could just say “Consider the set { {set of odds}, {set of evens} }.”  But that’s not a Category.

I would have to go on to define outside stuff, relate it to the inside stuff — it would be like bad object-oriented programming and it would certainly be hard to read.  With Categories the interpretation comes along for the ride, what-you-do-with-it is part of the what-it-is just like good OOP.

It’s almost Post-Modern.  Nothing comes without a context.  Things only have meaning within a context.  You have to bring the operator and the operated-on — the subject and the object — up at the same time.

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