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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!

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  • $\begingroup$ Have you thought about all this in the category of sets? $\endgroup$
    – Randall
    Jan 29 '20 at 1:39
  • $\begingroup$ Yes, it’s a bit fuzzy but I mostly understand how it works in Sets. My real issue is is when it comes to abstract categories. $\endgroup$
    – Owen
    Jan 29 '20 at 1:40
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    $\begingroup$ Especially with epimorphisms, you need to be careful about thinking of them as generalized surjective functions. For example, in the category of rings, the inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ is an epimorphism. Similarly, if $D$ is a dense subset of a topological space $X$ and you give $D$ the subspace topology, then the inclusion map $D \hookrightarrow X$ is an epimorphism in the category of topological spaces and continuous functions. $\endgroup$ Jan 29 '20 at 1:44
  • $\begingroup$ I suggest you to take one step at a time... Did they tell you what is "generalized" in their context? What is an abstract category for you? $\endgroup$ Jan 29 '20 at 3:23
  • $\begingroup$ @DanielSchepler I always thought that dense continuous functions being epi would require Hausdorff spaces... $\endgroup$
    – PrudiiArca
    Jan 29 '20 at 6:25
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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)...

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  • $\begingroup$ Awesome, thanks! That is in fact the exercise I’m doing for an assignment right now! $\endgroup$
    – Owen
    Jan 29 '20 at 12:52

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