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1. • CommentRowNumber1.
   • CommentAuthorJonasFrey
   • CommentTimeAug 4th 2015
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   Author: JonasFrey
   Format: MarkdownItexI 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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeAug 4th 2015
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   Author: Mike Shulman
   Format: MarkdownItex(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.

   (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.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 4th 2015
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   Author: DavidRoberts
   Format: MarkdownItexI 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorJonasFrey
   • CommentTimeAug 17th 2015
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   Author: JonasFrey
   Format: MarkdownItexWith 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](http://arxiv.org/pdf/math/0206203v2.pdf) (Corollaire 3.6), but I couldn't find a definition there!

   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!

5. • CommentRowNumber5.
   • CommentAuthorZhen Lin
   • CommentTimeAug 17th 2015
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   Author: Zhen Lin
   Format: MarkdownItexMy 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorJonasFrey
   • CommentTimeAug 17th 2015
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   Author: JonasFrey
   Format: MarkdownItexZhen 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 $f:A\to B$ 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 $B$. 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 $f:A\to B$ is initial in the category of parallel pairs that are equalized by $f$, so $f$ is the equalizer of its cokernel pair iff it is the equalizer of all the pairs that it equalizes.

   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 $f:A\to B$ 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 $B$. 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 $f:A\to B$ is initial in the category of parallel pairs that are equalized by $f$, so $f$ is the equalizer of its cokernel pair iff it is the equalizer of all the pairs that it equalizes.

7. • CommentRowNumber7.
   • CommentAuthorThomas Holder
   • CommentTimeAug 17th 2015
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   Author: Thomas Holder
   Format: MarkdownItexFor 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeAug 18th 2015
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   Author: Mike Shulman
   Format: MarkdownItex> 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.

   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.

9. • CommentRowNumber9.
   • CommentAuthorJonasFrey
   • CommentTimeAug 19th 2015
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   Author: JonasFrey
   Format: MarkdownItex> 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.

   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.