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I've been trying to get my head around the various types of epimorphisms you get in category theory, but I can't see why anyone uses "extremal" epis as opposed to the slightly less general notion of "strong" epis.

Every strong epi is extremal; extremal epis can be proved strong if you have pullbacks; so the notions coincide in pretty much any category you're likely to be working in. So for instance in Top the extremal epis = strong epis = quotient maps (as opposed to any old surjective continuous map).

What's more, the definition of a "strong" epi arises naturally when you try to work out what conditions you need to put on an epi to get unique epi-monic factorisation. Try proving that Set has unique epi-monic factorisation, for instance; you'll end up proving a lemma that states all epis in Set are strong.

The definition of "extremal", in contrast, seems to come out of nowhere. So why bother with extremals at all? Is there any use or motivation for the definition, or is it just some kind of historical hangover?

Thanks in advance for any light you can shed on this.

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I'd like to know the answer to this as well. I thought I'd seen this question before and then remembered, ah yes /r/math :) –  tharris Apr 13 '13 at 17:04
Keep in mind that even if you have a nice category $\mathbb{A}$ with pullbacks, its slice categories $A/\mathbb{A}$ can easily fail to have them. –  you-sir-33433 Oct 25 '13 at 9:03

3 Answers 3

Epimorhpisms are meant to capture the idea of 'surjectivity'. However, after some searching through concrete categories, one finds that epimorphisms don't quite get surjectivity right. They do in Top, Set, and Grp, but counter examples do creep up in Ring (rings with 1, and ring homomorphisms that preserve 1) such as the inclusion of the integers in the rationals. However, if we strengthened epimorphism to extremal epimorphism, we find these are precisely the surjective ring homomorphisms. In our counter-example above, the inclusion of the integers also factors through the inclusion of the dyadic rationals which is not an isomorpism (dyadic rationals are not a field). Furthermore, extremal epimorphisms also imply surjectivity in Haus where the epimorphisms are precisely the continuous maps with only dense images. Extremal epimorphisms still don't pin down surjectivity in all concrete categories, but they are a middle ground and imply surjectivity in the categories that are most studied.

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When $X \to I \to Y$ is an image factorization of a morphism $X \to Y$, then $X \to I$ does not factor through any proper subobject of $I$. Hence, if $X \to I$ is epi (which is quite typical), then it is an extremal epi. This is how they appear naturally, and perhaps it is one of their motivations. Strong epimorphisms are a little bit more complicated to define. I suspect that they are the correct analogue for categories where no pullbacks exist.

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Even if all you care about are strong epimorphisms, it's still nice to prove that they have nice properties, and one of those properties is that they're extremal.

I don't understand your claim that the definition of an extremal epimorphism comes out of nowhere. It seems like a natural condition to write down if you're thinking about epi-mono factorizations; an extremal epimorphism is one that doesn't factor through a nontrivial monomorphism.

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On the other hand, if you are trying to prove the uniqueness of epi–mono factorisations, you would need a lifting condition like that of a strong epimorphism. –  Zhen Lin Apr 13 '13 at 22:43

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