Morphism

A category consists of a set of objects and a set of arrows (along with the axioms). Morphisms or arrows connect objects, they are mappings between the objects in a category, connecting them in a directional way. Morphisms are the primary focus of category theory because they relate objects and from these relations we can infer many things about a category.

Morphisms

• the set of arrows in a category 𝒞 is denoted by Arr(𝒞)

• an arrow f going from object A to B is denoted by f : A → B

• an arrow f : A → B can also be denoted by f : Arr(A,B)

• that arrow f belongs to a category 𝒞 is denoted by f ∈ Arr(𝒞)

• identity arrow on object A is denoted by 1ᴀ : A → B or idᴀ : Arr(A,A)

Arrow symbols

• monomorphism (injective homomorphism) ↪, $\hookrightarrow$, ↣

• epimorphism (surjective homomorphism) ↠, $\twoheadrightarrow$

• isomorphism (bijective homomorphism) ≅ or ⥲, ⤐

Non-identity morphism

In a category 𝒞, if there is an arrow f : a → b, then we say that a is the source or domain object, and b is the target or destination or codomain object of the arrow f. This can be denoted by src (f) = dom(f) = a, tgt(f) = dest(f) = cod(f) = b.

In many categories arrows are functions, but not always. In 𝗥𝗲𝗹 category, objects are sets and arrows are relations between sets; e.g. <= is a morphism in a preorder category. In 𝗩𝗲𝗰𝘁, objects are vector spaces and arrows are linear transformations. In 𝗚𝗿𝗽, objects are groups, while the arrows are group homomorphisms.

Arrows themselves are also mappable - functors are a kind of higher-order arrows that map between morphisms.

Identity morphism

Each object A in a category 𝒞 must have an identity morphism that maps an object back to itself, idᴀ : A -> A. When understood as operations or processes, then an identity arrow signifies a "do nothing" process.

Identity arrows are denoted by id or 1 and may be subscripted with the object whose identity they represent, e.g. idꜰₐ, 1ᴀ and similar.

An identity arrow, on an object A, idᴀ : A → A, is not the same morphism as possible other arrows A ⟼ A. There could be many morphisms A ⟼ A, but only one of them is the identity of A. Just because two arrows have the same source and target object, doesn't mean they are the same. For example, there are a lot of arrows ℕ ⟼ ℕ (succ, factorial, square, etc.), but they are all distints from each other and from the identity arrow of ℕ.

[FAQ] However, in general, i.e. without a view into an object's internals, how are we to distinguish the id : A -> A from other arrows A ⟼ A?