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Category Theory

Objects, arrows, functors, universal properties: mathematics about mathematics.

Àwọn Àkọsílẹ̀

Advanced Categories, functors and natural transformations Objects and arrows, composition, and maps between categories. {1,2} union {3} Advanced Universal properties, limits and adjunctions Products, coproducts, limits and the adjunctions that unify constructions. lcm(4, 6)

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.

Àwọn ìwé-ìwé mìíràn

Objects and arrows

Category theory Category (mathematics)Morphism Isomorphism Functor Natural transformation Opposite category Subcategory Commutative diagram

Universal constructions

Universal property Product (category theory)Coproduct Limit (category theory)Pullback (category theory)Initial and terminal objects Adjoint functors Yoneda lemma Equivalence of categories

Structures

Monoidal category Monad (category theory)Abelian category Topos Higher category theory Homological algebra

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Àwọn Àtòjọ-ẹ̀yàn

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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