Monomorphism and epimorphism intuition

Question

I am new to category theory, and I am having trouble getting used to the idea of monos and epis.

I fully understand the definitions, but it seems like some of my classmates think of them as generalized versions of injectivity and surjectivity, and I’m having trouble understanding that, especially in abstract categories.

I’m also having a bit of trouble understanding the idea of a mono/epi which splits.

Thanks in advance!

Answer 1

In the category of sets there are many equivalent ways to characterize injective functions. A function $f:X\rightarrow Y$ is injective, iff for all functions $g_1,g_2:W\rightarrow X$ it holds that $f \circ g_1 = f \circ g_2$ implies $g_1 = g_2$, or iff there is a retract $h:Y\rightarrow X$ such that $h\circ f=id_X$ or finally if it has a certain universal property, which I dont know how to write here.

Note that all these equivalent formulations of injectivity are purely in terms of functions/arrows and no elements were required to state them. Hence we can state them in any arbitrary category.

The problem now is that, while in the category of sets all these notions are equivalent, in an arbitrary category they give rise to arrows with very different properties. For this reason thay come with distinct names, namely monomorphism, split monomorphism and regular monomorphism. The only implications that must hold in any category are that each split monomorphism is a regular monomorphism and that each regular monomorphism is a monomorphism.

You can do the same game with surjective functions, which will give you again notions of epimorhism, split epimorphism and regular epimorphism.

Hence your friends are right, when saying that monos/epis generalize injective/surjective functions, but be aware that there are multiple such generalizations.

PS a fun little exercise is to show that, while a morphism being mono + epi does not imply that it is an isomorphism (see $\mathbb Z \rightarrow \mathbb Q$), it is true that if it is mono and split epi (or split epi and mono)...

Answer 2

This confused me as well. Here is my understanding:

In category theory, we want to understand the behaviors of morphisms. We think of "surjectivity" as the idea that the image covers everything in the codomain. However, this idea turns out to be too restrict and the "correct" idea should be the image covers all information we need in terms of relations to other objects which is a superset of the previous one. For example as @DanielSchepler mentioned in the category of rings, $f:\mathbb{Z}\hookrightarrow\mathbb{Q}$ is a non-surjective epimorphism since any ring homomorphism from $\mathbb{Q}$ is determined by its value on $\mathbb{Z}$ i.e. for any ring $A$ and morphisms $g_1:\mathbb{Q}\to A, g_2:\mathbb{Q}\to A$, $g_1\circ f=g_2\circ f$ implies $g_1=g_2$. This exactly captures the idea that $f$ gives all the information you need. Similarly, injectivity means not losing any information in the domain but monomorphism means not losing any information needed in the domain. A non-injective monomorphic example is here.