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!

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