→ Category Theory

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Instead of asking what things are, ask how they map to each other. Category theory finds the same pattern (a product, a limit, an adjunction) recurring across every branch, and gives it one name.

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F: \mathcal{C} \to \mathcal{D}functor, natural transformation
A structure-preserving map between categories; a map between functors.\operatorname{Hom}(Aarrows from A to B, composition
The set of morphisms; do g then f.

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What is a functor?

A map between categories that sends objects to objects and arrows to arrows while respecting composition. Taking the fundamental group of a space is a functor from spaces to groups.

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