Category Theory

Abstraction

Aka general abstract nonsense. Abstract from sets and relations to categories and morphisms.

Category

(Class ob(C) of objects, morphisms or arrows hom(C), composition op: $\cdot$) with $\cdot$ identity, associative $\cdot$. Category Eg: Set, $Vec{t}_k$.

Small category: aka CAT: both ob(C) and hom(C) are sets, rather than classes.

Morphisms

Homomorphism: A structure (identity, inverse elements, and binary ops) preserving funciton f: f(x)=3x preserves addition. Isomorphism: both f and $f^{-1}$ are homomorphisms. Endomorphism: homomorphism of a mathematical object to itself. Automorphism is both isomorphism and an endomorphism.

Functors

Structure preserving mapping between categories and their morphisms.