hiremebecauseimsmart:

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:

111111111

I’ll write o for odd and e for even.

o+o=e
e+e=e
o+e=o
e+o=o

222222222

I’ll write P for positive, N for negative, and × for times.

N×N=P
P×P=P
N×P=N
P×N=N

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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 & even preserve the state of the thing they operate on, and negative & odd change the state.

Interchanging (e,o,+) for (P,N,×) is called a functor.

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

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