category theory

firebird
frobenius algebra
functors in haskell

Saunders Mac Lane, one of the founders of the field, actually argued that the concept of a monoid is the fundamental notion of category theory—seeing a category itself as a sort of "generalized monoid"

haskel

morphism
a category is just a list of morphisms
a morphism must compose and have an identity
functions more struct than morphisms
a matrix is a morphism but not a function

natural transformations are morphisms in the category of Func functors
not to be confused with category Cat where categories are objects and functors are the morphisms