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If I were the person creating category theory, I wouldn’t have been interested in left and right cancellation. I may or may not notice it, but it doesn’t feel fundamental enough for me to define it generally or give it a name. Question: if you tell me a morphism is epic or monic, what other insight do I gain about the morphism that is equivalent but conceptually what we actually use? (since just taking about cancellations seems unnatural and unworthy of a name to me right now).

As an example I’ve thought about stuff like “information is lost” but that doesn’t feel like it works somehow.

EDIT: ok I've heard they're supposed to be categorical generalisations of injections and bijections, but from the definition they don't always behave in ways you'd expect (just incase, I can find some places if you want). Is there a more general but relatable way to think about them that eliminates this occasional weirdness?

Pineapple Fish
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  • E.g. there is a strong correspondence between: monic, injective, subobject. Do concepts as "injective" and "subset/subgroup" also not feel fundamental enough for you to define? – drhab Jan 13 '19 at 16:52
  • Why is this being downvoted? I’m genuinely inquistive about what general thing makes them useful. – Pineapple Fish Jan 13 '19 at 20:27
  • You would have eventually come up with the notions, since they're motivated by elementary school arithmetic --to say the least ;-) See here: https://math.stackexchange.com/a/2885267/80406 for ``Epimorphism and monomorphism explained without math'' – Musa Al-hassy Jan 14 '19 at 16:18
  • @MusaAl-hassy Thanks :). I read that before I posted though. I know their definition, but why would I take elementary arithmetic into the generality of the universe. Surely we don't just define every random property of everything we notice, right? – Pineapple Fish Jan 16 '19 at 19:36
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    We model problems in the universe generally using formulae. For a large part, these formulae tend to be equations and contain unknowns. We wish to solve for such unknowns, and postulate desirable cancellation properties ;-)

    In the setting of relations, certain cancellable relations are called functions; others having more cancellation properties may be called surjective functions. Incidentally, relations give us inequational reasoning and functions shift that to equational.

    All the best :-)

     – Musa Al-hassy Jan 22 '19 at 12:43

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The concept of monomorphism is one of the most fundamental concepts of category theory. It is used to define posets of subobjects, which are used to define well-poweredness, which is one of the properties arising in the Special Adjoint Functor Theorem, one of the most fundamental theorems of category theory.

Monomorphisms and subobjects are deeply connected, as well as their duals epimorphisms and quotient objects, appearing almost in every topic of category theory. For example, in sheaf theory: the basic notion of a sieve can be defined as a subobject of a hom-functor, which may be presented as a monomorphism in the category of presheaves.

Monomorphisms and epimorphisms are generalizations of very important notions in different algebraic categories, what explains their wide usage in homological algebra. For example, in the theory of abelian categories, a monomorphism is the same as a kernel, which is also the very basic and fundamental notion.

So you can see that monics and epics are fundamental notions and are necessary to define in any "creation" of category theory. That's why you can find their definitions at the first pages of every category theory textbook.

Oskar
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    You can understand that something's important but not know how it actually works – Pineapple Fish Jan 16 '19 at 16:45
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    @BenjaminThoburn The point of my answer is that the importance of monics and epics is not in the cancellation itself, but in their value in category theory and all mathematics. Concepts are added to category theory not taking into account their abstract expressions, but because of their importance in examples, applications (categorical, mathematical, physical etc) and because of their good properties. Cancellation is only the simplest way to define monics and epics; there are many other definitions (especially in categories with some good properties). – Oskar Jan 17 '19 at 00:16