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The duality between monomorphisms and epimorphisms is well understood in terms of the opposite category (reversing the arrows). However, it is much less obvious on intuitive level. To cite Notes on Category Theory

  • Intuitively, a monomorphism is a map which “does not map different things of X to the same thing in Y”. (p22)

  • An epimorphism, intuitively, is “something having full image” (p25)

These are not quite dual statements at all. I suspect that the culprit of this skewed duality might have something to do with the map nature of morphisms, and this intuition would simply fail for relations (as opposed to functions). Are there little more satisfactory explanations?

Tegiri Nenashi
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    You might think about what these mean in $\textbf{Set}$ where you can rephrase a monomorphism as $f(x) = y$ has at most one solution and an epimorphism as $f(x) = y$ has at least one solution. The idea is similar, although of course general morphisms aren't really maps anyway. – Osama Ghani Apr 05 '21 at 20:46
  • @OsamaGhani It is also remarkable that the first order sentences "at most one solution" and "at least one solution" are not exactly dual from logic perspective. – Tegiri Nenashi Apr 05 '21 at 21:24
  • I would say that those "intuitions" that you quote are unhelpful. See https://math.stackexchange.com/questions/2075543/a-monomorphism-that-is-not-injective and https://math.stackexchange.com/questions/81123/examples-of-categories-where-epimorphism-does-not-have-a-right-inverse-not-surj for relevant examples. These abstractions are about how some morphisms "control" other morphisms, e.g., a morphism of $\Bbb{Q}$ into another ring is controlled by its restriction to $\Bbb{Q}$. – Rob Arthan Apr 05 '21 at 23:20
  • ... I was too slow trying to fix the error at the end of my comment: $\Bbb{Q}$ at the end should read $\Bbb{Z}$. – Rob Arthan Apr 05 '21 at 23:32
  • Even in $\mathsf{Set}$, the duality breaks down. The equivalence of “$f\colon X\to Y$ is a monomorphism” and “$f$ has a left inverse” only requires $X\neq\varnothing$. By contrast, the equivalence of “$f\colon X\to Y$ is an epimorphism” and “$f$ has a right inverse” is equivalent to the Axiom of Choice. – Arturo Magidin Apr 06 '21 at 04:05
  • I wouldn't say dual objects should have similar or opposite behaviour which is what your intuition might hinge on. Similarly, categorically, injective and projective objects of a category are dual but they behave quite differently. (So some people even say that flat covers are the dual of injective envelopes, not projective covers. But I'm not so sure if I would agree with that.) – Qi Zhu Apr 06 '21 at 14:43

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