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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 2nd 2010

felt like adding a handful of basic properties to epimorphism

• CommentRowNumber2.
• CommentAuthorJohn Baez
• CommentTimeOct 28th 2020

Noted that any functor that preserves pushouts preserves epimorphisms.

• CommentRowNumber3.
• CommentAuthorGuest
• CommentTimeDec 10th 2020
Below the proof of Proposition 4.7, it is noted that

> 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>
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeDec 10th 2020

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)

1. 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

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJan 22nd 2021

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.)

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeMar 18th 2021

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.

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeMar 18th 2021

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.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeMar 19th 2021

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).