Composition of Monomorphisms and Epimorphisms

In this post, I want to discuss a very basic property of monomorphisms and epimorphisms. So to begin, I will start by stating each of their formal definitions taken closely verbatim from Aluffi’s Algebra: Chapter $0$ text:

Definition. (Monomorphism) Let $\mathcal{C}$ be a category. A morphism $f\in\text{Hom}_\mathcal{C}(A,B)$ is a monomorphism $\iff$ $\forall$ objects $Z \ \text{of} \ \mathcal{C}$ and all morphisms $\alpha',\alpha''\in\text{Hom}_\mathcal{C}(Z,A),$

$f\circ\alpha'=f\circ\alpha''\implies\alpha'=\alpha''.$

Definition. (Epimorphism) Let $\mathcal{C}$ be a category. A morphism $f\in\text{Hom}_\mathcal{C}(A,B)$ is a monomorphism $\iff$ $\forall$ objects $Z \ \text{of} \ \mathcal{C}$ and all morphisms $\beta',\beta''\in\text{Hom}_\mathcal{C}(B,Z),$

$\beta'\circ{f}=\beta''\circ{f}\implies\beta'=\beta''.$

From here, we may proceed in trying to prove the composition of monomorphisms and epimorphisms. We start with the composition of monomorphisms:

Let $f,g\in \text{Obj}(\mathcal{C})$, and $f:\text{Hom}_\mathcal{C}(A,B), g:\text{Hom}_\mathcal{C}(B,C)$ be monomorphisms. We take the composition of $f$ and $g$ to be $fg:\text{Hom}_\mathcal{C}(A,C)$. From this we may show that by the of definition a  monomorphism, $fg$ is also a monomorphism when both when both $f$ and $g$ are monomorphisms themselves. So,

$(f\circ{g})\circ\alpha'=$

$[\text{Hom}_\mathcal{C}(A,B)\circ\text{Hom}_\mathcal{C}(B,C)]\circ\text{Hom}_\mathcal{C}(Z,A)=$

$\text{Hom}_\mathcal{C}(A,C)\circ\text{Hom}_\mathcal{C}(Z,A)=$

$\text{Hom}_\mathcal{C}(Z,C)=$

$(f\circ{g})\circ\alpha''\implies\alpha'=\alpha''.$

Q.E.D.

Similarly, we prove the composition of epimorphisms very close to the proof for the monomorphic case excluding the assumptions that were already made:

We show that by the of definition an epimorphism, $fg$ is also a epimorphism when both when both $f$ and $g$ are epimorphisms themselves. So,

$\beta'\circ(f\circ{g})=$

$\text{Hom}_\mathcal{C}(B,Z)\circ[\text{Hom}_\mathcal{C}(A,B)\circ\text{Hom}_\mathcal{C}(B,C)]=$

$\text{Hom}_\mathcal{C}(B,Z)\circ\text{Hom}_\mathcal{C}(A,C)$

$\text{Hom}_\mathcal{C}(A,Z)=$

$\beta''\circ(f\circ{g})\implies\beta'=\beta''.$

Q.E.D.

There will be further discussion and proofs on category theory and some of the current research that I have finally started!

6 thoughts on “Composition of Monomorphisms and Epimorphisms”

1. Joshua Davidson

What are the monomorphisms and epimorphisms in the category of sets?

Like

• Julian Rachman

To be put short, monomorphisms and epimorphisms in the category of sets are usually represented as injective (one-to-one) functions between sets following that if $a'\neq{a''}\implies{f(a')}\neq{f(a'')}.$

Like

2. David