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felt like adding a handful of basic properties to epimorphism
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)
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.)
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.
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.
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).
Corrected the type of the canonical map $i_1$ in the proof that the coproduct of epimorphisms is an epimorphism: the map was given as $i_1 : x_1 \to x_1 + x_2$ but it has to be $i_1 : y_1 \to y_1 + y_2$ for $g \circ i_1 \circ f_1$ to make sense when $f_1 : x_1 \to y_1$ and $g : y_1 + y_2 \to z$.
Anonymous
Added:
added pointer to:
Changed “strong epimorphism = an epimorphism right orthogonal to monomorphisms” to left orthogonal (to match strong epimorphism).
Added simpler example of a ring epimorphism that’s not surjective. Reordered text so that “But beware that the converse fails:” comes after the statement it’s supposed to, namely “Every isomorphism is both an epimorphism and a monomorphism.”
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