Les Categories Pour Les Nuls

Math Online Tom Circle

“Categories for Dummies”
(French)

Example 1:

Paris (P) -> Rome (R) -> Amsterdam (A)
Objects: cities {P, R , A}
Morphism (Arrow): railway

  • Identity: railway within the city
  • Associative: (P -> R) -> A = P -> (R ->A)

=> Category

Example 2:
A, B are categories

functor f : A -> B

f (B) has the “information” on A, with some loss of information since f may not be a MONOMORPHISM.

Example 3: Natural Transformation

A = 0 1 2 3 4
f : A -> B

B = Ladder steps:
f(0)|
f(1)|
f(2)|
f(3)|

g : A -> B

B = Staircase steps :
g(0)||
g(1)||
g(2)||
g(3)||

Natural Transformation: =>α

α : f (i) => g (i)

f(0)| =>g(0)||
f(1)|=>g(1)||
f(2)|=>g(2)||
f(3)|=>g(3)||

α transforms naturally the Ladder to the Staircase.

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About tomcircle

Math amateur
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