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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.

    diff, v30, current

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

    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)

    diff, v36, current

  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

    diff, v38, current

    • 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.)

    diff, v41, current

    • 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).

  2. Corrected the type of the canonical map i 1i_1 in the proof that the coproduct of epimorphisms is an epimorphism: the map was given as i 1:x 1x 1+x 2i_1 : x_1 \to x_1 + x_2 but it has to be i 1:y 1y 1+y 2i_1 : y_1 \to y_1 + y_2 for gi 1f 1g \circ i_1 \circ f_1 to make sense when f 1:x 1y 1f_1 : x_1 \to y_1 and g:y 1+y 2zg : y_1 + y_2 \to z.

    • Jussi Piitulainen


    diff, v43, current

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 22nd 2023


    • G. M. Kelly, Monomorphisms, Epimorphisms, and Pullbacks, Journal of the Australian Mathematical Society 9.1-2 (1969), pp. 124–142. doi:10.1017/s1446788700005693](

    diff, v46, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023

    added pointer to:

    diff, v48, current

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeNov 11th 2023

    Extended and started more precise the discussion of an epimorphism of rings. Still under construction though (written in a bus, and will be checked and improved later).

    Added an example of epimorphism of sheaves of sets over a topological space.

    diff, v50, current

    • CommentRowNumber14.
    • CommentAuthorʇɐ
    • CommentTimeJan 1st 2024

    Changed “strong epimorphism = an epimorphism right orthogonal to monomorphisms” to left orthogonal (to match strong epimorphism).

    diff, v52, current

    • CommentRowNumber15.
    • CommentAuthorJohn Baez
    • CommentTimeJan 23rd 2024

    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.”

    diff, v53, current