After learning Haskell I finally decided to jump the train and start learning Category Theory with the help of the book Conceptual Mathematics – A first introduction to categories. I greatly encourage you to do the same, especially if you are a self-taught programmer.
The following is a simple cheat sheet that is meant to help me (and maybe you?) remember all those new words and constructs. Please poke me if you find any errors or have an improvement suggestion.
Basics
Associative
The order of operator application does not matter.
.
Commutative
The order of the operands does not matter.
.
Idempotent [idem + potence = (same + power)]
The endomorphism 
.
Involution
The automorphism 
.
Category
What makes a Category?
– Objects
– Morphisms (maps)
– For each object in 
– For each pair of maps 
– Composition 
.
Morphisms
Monomorphism (injective) [monos = one]
– distinctive
It maps distinct objects onto distinct objects. The codomain is at least as large as the domain.
– left cancellative
.
Split Monomorphism
A monomorphism 
.
Epimorphism (surjective) [epis = against]
– covering
All objects of the codomain are hit at least once. The domain is at least as large as the codomain.
– right cancellative
.
Split Epimorphism
An epimorphism 
.
Isomorphism (bijective) [isos = equal]
– is a monomorphism and epimorphism
– 
– distinctive
– invertible
.
Automorphism [auto = self]
– is an isomorphism
– also called permutation
.
Homomorphism [homos = same]
– structure preserving map
– if 
.
Endomorphism [endos = inside]
– is a homomorphism
.
Constant map
A map
.
Division of maps
Determination or extension
If it has a solution g, we say h is ‘determined by’ f or h ‘depends only on’ f.
.
Retraction
Special case of the determination problem 
– is an epimorphism
– left inverse of
.
Choice or lifting
.
Section
Special case of the choice problem 
– is a monomorphism
– right inverse of
.
