## How jectivity corresponds to morphisms

“Jectivity” may or may not be a real word in the dictionary and my command of the English language is not adequate for me to make such a discussion.

But “jectivity” is indeed a valid mathematical word and it concerns the classification of functions into injective, surjective, and bijective ones.

But before we discuss jectivity, let us discuss functions. What is a function and what is not a function?

A function maps all elements from its domain to its codomain. Any element from the domain is mapped to only one element from the codomain.

This is a function:

This is not a function, because an element from the domain is not mapped:

This is not a function, because an element from the domain is mapped to more than one element in the codomain:

Now let us classify functions into injective, surjective, and bijective ones.

Please note that the following analysis is based on numerous Wikipedia articles on functions and morphisms.

An injective function (or injection) maps any element of the domain to a different element in the codomain.

A surjective function (or surjection) maps to all elements of the codomain.

A bijective function (or bijection) is both injective and surjective.

Example:

A function f: A -> B is injective if and only if f is left-invertible; that is, there is a function g: f(A) -> A such that g o f = identity function on A.
A function f: X -> Y is injective if and only if it is left-cancellative; that is, given any functions g1 ,g2 : Y -> Z, f o g1 = f o g2 => g1 = g2.

A function f: A -> B is surjective if and only if it is right-invertible; that is, there is a function g: B -> A such that f o g = identity function on B.
A function f: X -> Y is surjective if and only if it is right-cancellative; that is, given any functions g1, g2 : Y -> Z, g1 o f = g2 o f => g1 = g2.

A function f: A -> B is bijective if and only if it is invertible; that is, there is a function g: B -> A such that g o f = identity function on A and f o g = identity function on B.

The composition of two bijections is again a bijection, but if g o f is a bijection, then it can only be concluded that f is injective and g is surjective.

Injections, surjections, and bijections loosely correspond to Category Theory’s monomorphisms, epimorphisms, and isomorphisms, respectively.

A monomorphism (also called a monic morphism or a mono) is a morphism f: X -> Y that is left-cancellative in the sense that, for all morphisms g1, g2 : Z -> X, f o g1 = f o g2 => g1 = g2.

An epimorphism (also called an epic morphism or an epi) is a morphism f: X -> Y that is right-cancellative in the sense that, for all morphisms g1, g2 : Y -> Z, g1 o f = g2 o f => g1 = g2.

An isomorphism is both a monomorphism and an epimorphism.

A morphism f : X -> Y in a category is an isomorphism if it has a two-sided inverse; that is, there is another morphism g : Y -> X in that category such that g o f = 1X and f o g = 1Y, where 1X and 1Y are the identity morphisms of X and Y, respectively.

I think that this analysis could be used as the introduction to a primer in Category Theory.