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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2017
    • (edited May 7th 2017)

    The entry monomorphism used to start off saying that a monomorphism is an epimorphism in the opposite category…

    I have polished and expanded the text now, trying to make it look more like an actual exposition and explanation. I have also expanded a little the Examples-section, and similarly at epimorphism.

    These weird kind of entries date from the early days of the nnLab, when none of us saw yet what the nnLab would once be. Back then it was fun to proceed this way, now it feels awkward.

    I hereby pose a challenge to the nnForum community:

    I challenge you to each pick one entry on a basic topic (nothing fancy), go to the corresponding nnLab entry and give it a gentle introductory Idea-section, make sure that the basic motivating examples are mentioned in the order in which the newbie needs to see them, and that the key facts are stated as nicely discernible propositions, best with proof or at least with some helpful pointer, in short, to make the entry a useful read for those readers who would profit from reading it, especially those who do not know the nPOV yet, but might be guided to learn and appreciate it.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 7th 2017

    Great idea, Urs.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 7th 2017

    I agree, good idea. There is so much work to do here.

    I find “dual of epimorphism” pretty funny. If there’s an order to these concepts, it’s arguably in the other direction: limit notions should come before colimit notions. What I mean is that f:ABf: A \to B is epic iff hom(f,X)\hom(f, X) is monic in SetSet for all XX, and similarly all limits and colimits in categories reduce to limits in SetSet. Second, for the classical large categories of structures, epimorphisms are harder to understand than monomorphisms.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2017
    • (edited May 8th 2017)

    While proof-reading monomorphism, I see we find ’monic’ used unannounced. I could just replace them all by ’monomorphism’, but how widely used is ’monic’?

    Also epimorphism has ’epic’ used (once) unexplained. And ’epi’. Strangely, both seemed to be used as adjectives, where all ’monic’s are nouns.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2017

    David, please add a line explaining the jargon.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 8th 2017

    I use ’monic’ and ’epic’ all the time as adjectives, and ’mono’ and ’epi’ as nouns. I thought the practice was widespread.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2017
    • (edited May 8th 2017)

    So, Todd, would you have ’monos’ instead here, or is there ellipsis for ’monic arrows’?

    Frequently, regular and strong monics coincide.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 8th 2017

    Yes, it should be ’monos’, and I say this even if I was the one who wrote that.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2017

    Ok, I think both entries are consistent now.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2017

    In the spirit of the challenge in #1, I did add a line explaining the jargon to the Idea-sections at monomorphism, epimorphism and isomorphism.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2017

    I had added something to that effect later, there and at epimorphism:

    A morphism f:XYf \colon X \to Y in some category is called a monomorphism (sometimes abbrieviated to mono), or described as being monic, if …

    But maybe better earlier where you have it.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2017

    Oh, I didn’t see that, sorry. But it doesn’t hurt to say it in the Idea-section already.

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeMay 8th 2017
    • (edited May 8th 2017)

    Entry says

    Since injective functions are precisely the monomorphisms in Set (example \ref{MonomorphismsInSet} below) this may be stated as saying that ff is a monomorphism if for all objects ZZ then Hom(Z,f)Hom(Z,f) is a monomorphism.

    The final “then” feels awkward to my feeling of English, but I am not going to correct it as many native speakers work around. Even logically we can do the analysis. I mean why implication between the quantifier and the clause. “If” is complemented by the preceeding clause “ff is a monomorphism”.

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 8th 2017

    I agree with you, Zoran.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2017
    • (edited May 8th 2017)

    I am doing this intentionally to avoid two mathematical formulas appearing in a sentence in the role of two consecutive words. (Even with a comma separating them, this is awkward). I do seem to recall that another native English speaker once active here used to do this, too. But if you tell me that I must be misremembering, then I’ll believe you.

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 8th 2017
    • (edited May 8th 2017)

    Ah, I see. I think about this point (about adjacent formulas) a lot too. Let me rearrange. (Done.)

    • CommentRowNumber17.
    • CommentAuthorTim_Porter
    • CommentTimeMay 9th 2017

    That works well now. There are lots of places in papers written in English (and sometimes with native English speakers as authors!) where a very slight change in wording / word order can make a sentence much easier to read, even to parse for its intended meaning. There are questions of personal preference here, even of ’taste’, but, for example, starting a sentence with ’Then’ rarely works well in my view.

    • CommentRowNumber18.
    • CommentAuthorJohn Baez
    • CommentTimeOct 28th 2020

    Added an extra property of monos (now the first one) to match the page on epis.

    diff, v28, current

    • CommentRowNumber19.
    • CommentAuthorHurkyl
    • CommentTimeOct 28th 2020

    Since you still have the lock… you forgot to change “pushout” to “pullback” in proposition 4.1.

    • CommentRowNumber20.
    • CommentAuthorHurkyl
    • CommentTimeOct 28th 2020

    Also, if I haven’t gotten turned around, the characterization of epimorphisms and monomorphisms via yoneda lemma is backwards. I.e. the monomorphism page has the version characterizing epis, and the epimorphism page has the version characterizing monos.

    • CommentRowNumber21.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 20th 2020

    Fixed the errors noted by Hurkyl.

    diff, v33, current

    • CommentRowNumber22.
    • 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

    diff, v34, current

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