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.

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### Commutative

The order of the operands does not matter.

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### Idempotent [idem + potence = (same + power)]

The endomorphism is idempotent under composition if

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### Involution

The automorphism is an involution of A if

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# 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:

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# 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

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#### Split Monomorphism

A monomorphism which has a left inverse for which , i.e. a retraction.

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### 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

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#### Split Epimorphism

An epimorphism which has a right inverse for which , i.e. a section.

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**Isomorphism** (bijective) [isos = equal]

– is a monomorphism and epimorphism

– domain and codomain have the ‘same size’

– distinctive

– invertible

with and

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**Automorphism** [auto = self]

– is an isomorphism

– also called permutation

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### Homomorphism [homos = same]

– structure preserving map

– if and are groups, then f is a homomorphism if

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### Endomorphism [endos = inside]

– is a homomorphism

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### Constant map

A map is called constant if all elements of A are send onto the same element of B.

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# 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.

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#### Retraction

Special case of the determination problem in which and

– is an epimorphism

– left inverse of

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### Choice or lifting

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#### Section

Special case of the choice problem in which and

– is a monomorphism

– right inverse of

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