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    • CommentRowNumber1.
    • CommentAuthorJonasFrey
    • CommentTimeAug 4th 2015

    I came across a non-standard definition of “regular monomorphism” in Cassidy/Hébert/Kelly’s “Reflective subcategories, localizations and factorizations systems.” and added a note to the nlab page. They define a regular mono to be a joint equalizer of an arbitrary family of parallel pairs. This is more general than the usual definition, and forces the class of regular monos to be closed under arbitrary intersections.

    I think that in a well powered category with small products the definition should coincide with the usual one, and in coregular categories both should coincide with “strong mono”.

    Any comments? Does this definition of regular mono appear anywhere else? Or is there maybe an alternative term for it?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeAug 4th 2015

    (Side note: it’s helpful to always provide a link to the page you’re talking about, here regular monomorphism.)

    I think on the nLab we’ve called the dual notion a strict epimorphism. Which I don’t think is a particularly good name, but at least it disambiguates.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 4th 2015

    I thought that too, Mike, about strict epis (which I think is due to Grothendieck; I learned it from a paper of Dubuc on Grothendieck Galois theory), but I wasn’t sure.

    • CommentRowNumber4.
    • CommentAuthorJonasFrey
    • CommentTimeAug 17th 2015

    With a bit of delay – thank you Mike and David! Yes, strict monormorhpism seems to be the established term (although “Isbell, Structure of Categories” uses strict subobject to mean joint equalizer of a small – not arbitrary – set of parallel pairs). Regarding the origin and the link to Grothendieck: épimorphisme strict is used in SGA1 (Corollaire 3.6), but I couldn’t find a definition there!

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeAug 17th 2015

    My impression is that a strict monomorphism are joint equaliser of the collection of all parallel pairs it equalises, whereas the notion under discussion is only the joint equaliser of some parallel pairs.

    • CommentRowNumber6.
    • CommentAuthorJonasFrey
    • CommentTimeAug 17th 2015

    Zhen Lin, I don’t think this is a contradiction. Every joint equalizer is also the equalizer of all the pairs that it equalizes. So a morphism f:ABf:A\to B is a joint equalizer of all the pairs it equalizes iff it is the joint equalizer of an arbitrary of parallel pairs with joint domain BB. If anything, your comment is an argument in favor of not having any smallness restriction on the size of the family, since only then we have this equivalence in general.

    Another way of understanding strict monos is to say that it’s the class of regular monos closed under arbitrary existing intersections.

    In presence of cokernel pairs I think that strict monos coincide with regular monos, since the cokernel pair of f:ABf:A\to B is initial in the category of parallel pairs that are equalized by ff, so ff is the equalizer of its cokernel pair iff it is the equalizer of all the pairs that it equalizes.

    • CommentRowNumber7.
    • CommentAuthorThomas Holder
    • CommentTimeAug 17th 2015

    For the epimorphism concept, it might be helpful to consult the index of SGA4 (LNM 269) and Börger, Tholen, Strong regular and dense generators , in 1991 Cahiers. For the monomorphism concept, there might be something in Street, The family approach to total cocompleteness and toposes , transactions of the AMS 1984.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeAug 18th 2015

    Another way of understanding strict monos is to say that it’s the class of regular monos closed under arbitrary existing intersections.

    Isn’t that only true if equalizers of single parallel pairs exist? Otherwise there might be a mono that is a “joint equalizer” of lots of things but the individual equalizers of each of those pairs may not exist.

    • CommentRowNumber9.
    • CommentAuthorJonasFrey
    • CommentTimeAug 19th 2015

    Isn’t that only true if equalizers of single parallel pairs exist? Otherwise there might be a mono that is a “joint equalizer” of lots of things but the individual equalizers of each of those pairs may not exist.

    Indeed, good point.