Category Theory
Yoneda Lemma
Natural transformations from a representable functor y_X to F biject with elements of F(X)
Adjoint Functors
A pair of functors L ⊣ R with a natural bijection Hom(LX, Y) ≅ Hom(X, RY)
Limits and Colimits
Universal cones over diagrams: limits generalize products and equalizers; colimits generalize coproducts and coequalizers
Natural Transformation
A family of morphisms α_X : F(X) → G(X) making every naturality square commute
Functor Composition
Functors compose associatively with identity functors as units, forming the category Cat
Representable Functor
A functor F is representable if it is naturally isomorphic to Hom(X,-) for some object X
Products and Coproducts
Binary limits and colimits: products with projections and coproducts with injections satisfy dual universal properties
Monad
A monad is a functor T with unit η and multiplication μ satisfying associativity and unit coherence