Category :: Definition

A category 𝒞 consists of 3 components:

• a set of objects Obj(𝒞)

• a set of arrows Arr(𝒞)

• a set of axioms

Axioms (structural conditions):

1. Identity

   All objects must have an identity arrow.

each object A in category C must have an identity arrow, 1ᴀ

for all objects A there exists an identity morphism 1ᴀ such that for all morphisms f : A → B we have f ◦ 1ᴀ = f = 1ʙ ◦ f

∀X. X ∈ Obj(𝒞). f ◦ 1ᴀ = f = 1ʙ ◦ f

• Morphisms are closed under composition: if there is a morphism f : A → B and a morphism g : B → C, then there must be a morphism obtained by the composition, g ◦ f : A → C.

• Composition is associative: f ◦ (g ◦ h) = (f ◦ g) ◦ h

Common notation

• generic category : 𝒞 𝒟

• concrete category : 𝗦𝗲𝘁 𝗚𝗿𝗽 𝗛𝗮𝘀𝗸

• generic objects : A B C

• generic arrows : f g h

• generic functors : F G

• an arrow from A to B: Arr(A, B), F : A -> B

• identity arrow (object-subscripted): idᴀ 1ᴀ idꜰ₍ᵦ₎ idꜰᴀ

• all object in a category C: ∀X. X ∈ Obj(C)

• all arrows in a category C: ∀f. f ∈ Arr(C)

• Functor F, 𝓕, G

• f ◦ 1ᴀ = f

• f ◦ 1ᴀ = f = 1ʙ ◦ f

• ◦ =

Definition 2

A category C is given by

• a collection C₀ of objects

• a collection C₁ of arrows

  which have the following structure:

• each arrow has an object as source and target, f: A -> B

• identity: each object has identity arrow, 1ₐ or Iₐ

• transitivity axiom: if there is an arrow a -> b and an arrow b -> c then there must be an arrow a -> c

Definition 3

A category C is an algebraic structure consisting of a class of objects (denoted by A, B, C, etc.), and a class of arrows (denoted by f, g, h, etc.), together with three total and one partial operation.

The first two total operations are called target and source; both assign an object to an arrow. We write f : A <- B (pronounced "f has type A from B") to indicate that the target of the arrow f is A and the source is B.

The third total operation takes an object A to an arrow idᴀ : A <- A, called the identity arrow on A.

The partial operation is called composition and takes two arrows to another one. The composition f ∘ g (read "f after g") is defined iff f : A <- B and g : B <- C for some objects A, B, C, in which case f ∘ g : A <- C. In other words, if the source of f is the target of g, then f ∘ g is an arrow whose target is the target of f and whose source is the source of g.

Composition is required to be associative and to have identity arrows as units:

∀fgh. (f: A <- B) ∧ (g: B <- C) ∧ (h: C <- D) . f ∘ (g ∘ h) = (f ∘ g) ∘ h = f ∘ g ∘ h

∀f. f: A <- B . idᴀ ∘ f = f = f ∘ idʙ

Definition 4

A category 𝒞 consists of:

• objects, ∀A ∈ Obj(𝒞)

• arrows, ∀f ∈ Arr(𝒞)

An arrow f from an object A to B is denoted f : A -> B and

• the source object (domain) : dom(f) = A

• the target object (codomain): cod(f) = B

• For each object 𝑎, an identity arrow id𝑎 ∶ 𝑎 → 𝑎

• For each pair of arrows 𝑓 ∶ 𝑎 → 𝑏 and 𝑔 ∶ 𝑏 → 𝑐, a composite arrow 𝑔 ∘ 𝑓 ∶ 𝑎 → 𝑐. That is, for each pair of arrows 𝑓 and 𝑔 with cod(𝑓) = dom(𝑔), a composite arrow 𝑔 ∘ 𝑓 ∶ dom(𝑓) → cod(𝑔)