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 text:
Definition. (Monomorphism) Let be a category. A morphism is a monomorphism objects and all morphisms
Definition. (Epimorphism) Let be a category. A morphism is a monomorphism objects and all morphisms
From here, we may proceed in trying to prove the composition of monomorphisms and epimorphisms. We start with the composition of monomorphisms:
Let , and be monomorphisms. We take the composition of and to be . From this we may show that by the of definition a monomorphism, is also a monomorphism when both when both and are monomorphisms themselves. So,
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, is also a epimorphism when both when both and are epimorphisms themselves. So,
There will be further discussion and proofs on category theory and some of the current research that I have finally started!