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 is idempotent under composition if
.
Involution
The automorphism is an involution of A if
.
Category
What makes a Category?
– Objects
– Morphisms (maps)
– For each object in an identity map
must exist.
– For each pair of maps and
a composite map
must exist.
– Composition must follow the identity and associative laws:
.
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
with
.
Split Monomorphism
A monomorphism which has a left inverse
for which
, i.e. a retraction.
.
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
with
.
Split Epimorphism
An epimorphism which has a right inverse
for which
, i.e. a section.
.
Isomorphism (bijective) [isos = equal]
– is a monomorphism and epimorphism
– domain and codomain have the ‘same size’
– distinctive
– invertible
with
and
.
Automorphism [auto = self]
– is an isomorphism
– also called permutation
.
Homomorphism [homos = same]
– structure preserving map
– if and
are groups, then f is a homomorphism if
.
Endomorphism [endos = inside]
– is a homomorphism
.
Constant map
A map is called constant if all elements of A are send onto the same element of B.
.
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 in which
and
– is an epimorphism
– left inverse of
.
Choice or lifting
.
Section
Special case of the choice problem in which
and
– is a monomorphism
– right inverse of
.
Hey Paul, you might find this Category Cheat Sheet helpful; let me know what you think!
Thank you :-) Late reply by me. What an useful find!