Oh, thanks David this is quite helpful! There is no complete answer there, and I did read in detail the second (earlier, 1999) listed paper around 2006 or so, and it flashed across my vague memory when I was yesterday asking the question, but did not look at it as I felt it was not what I was looking for. Now I see it does have much of relevant reasoning on most important classes of epimorphisms and factorizations of interest. I will have some use of it (in fact it is also relevant for some of my interest in so called iterated localizations and 2-categorical analogues).

]]>Try this: *Epimorphic functors* and this: *Generalized congruences – Epimorphisms in Cat* (both by the same people, I strongly suspect there is a lot of overlap, but I haven’t checked.

Just occured to me, an elementary question, which functors are epimorphisms in the 1-category of small categories ? It is clear that all localization functors belong to this class.

]]>Gave the (counter-)examples of monos that are epi but not iso their own sub-section (here) and added mentioning of the example of dense subspace inclusions in Hausdorff spaces.

(Also cleaned up the previous material, as the example $\mathbb{Z} \hookrightarrow \mathbb{Q}$ had been mentioned twice.)

]]>It said that any morphism from an initial object is an epimorphism, which is not true. I changed it to say that any morphism TO an initial object is an epimorphism, which is the proof actually shows anyway.

Jonathan Beardsley

]]>Thanks for the alert.

This statement was introduced in rev 31. Maybe what was really meant was the converse statement, that an epimorphism implies a global section only in well-pointed toposes.

For the moment I have just removed the corresponding half-sentence. But everyone please feel invited to (re-)expand on this point.

(NB, the paragraph in question is here)

]]>> For example, the unique morphism to the terminal object is not always an epimorphism (though this is true in Set)

I believe the unique function from the empty set to a singleton is not surjective, so the claim is false in Set as well.

- Aniruddh Agarwal <me@anrddh.me> ]]>

Noted that any functor that preserves pushouts preserves epimorphisms.

]]>felt like adding a handful of basic properties to epimorphism

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