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1. • CommentRowNumber1.
   • CommentAuthorRodMcGuire
   • CommentTimeSep 1st 2017
   • (edited Sep 1st 2017)
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI've added the section [[relation#the_quasitopos_of_endorelations]] I was unsure whether to add this to [[relation]] or [[Quiv]] and somewhere we should explicitly give the [[subobject classifier]] of [[Quiv]]. Do people dislike my terminology or approach? Does $EndoRel$ have slick sub categories? Does it need a translation such as found at [[quality type#quality_types_as_localizations]] example 4.3 > Let $\mathbf{Bin}$ be the category of sets equipped with a binary relation i.e. objects are pairs $(X,\rho)$ with $X$ a set and $\rho$ a binary relation on $X$ and morphims $(X_1,\rho_1)\to (X_2,\rho_2)$ are functions $f:X_1\to X_2$ such that $x\rho_1 y$ implies $f(x)\rho_2 f(y)$. This is the same as the category of [[quiver|simple directed graphs]] hence a [[quasitopos]] since it corresponds to the separated objects for the [[double negation|double negation topology]] on the directed graphs.

   I’ve added the section relation#the_quasitopos_of_endorelations

   I was unsure whether to add this to relation or Quiv and somewhere we should explicitly give the subobject classifier of Quiv.

   Do people dislike my terminology or approach? Does $EndoRel$ have slick sub categories?

   Does it need a translation such as found at quality type#quality_types_as_localizations example 4.3

   Let $\mathbf{Bin}$ be the category of sets equipped with a binary relation i.e. objects are pairs $(X,\rho)$ with $X$ a set and $\rho$ a binary relation on $X$ and morphims $(X_1,\rho_1)\to (X_2,\rho_2)$ are functions $f:X_1\to X_2$ such that $x\rho_1 y$ implies $f(x)\rho_2 f(y)$. This is the same as the category of simple directed graphs hence a quasitopos since it corresponds to the separated objects for the double negation topology on the directed graphs.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 1st 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI think it's fine, but it should be cross-linked more (btw the same observation has also been made at [[quasitopos]]). I can add the description of the subobject classifier of $Quiv$ at [[Quiv]] if this is desired (it's not a hard calculation).

   I think it’s fine, but it should be cross-linked more (btw the same observation has also been made at quasitopos). I can add the description of the subobject classifier of $Quiv$ at Quiv if this is desired (it’s not a hard calculation).

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 2nd 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSo I added a bit to [[Quiv]], including a rough picture of its subobject classifier.

   So I added a bit to Quiv, including a rough picture of its subobject classifier.

4. • CommentRowNumber4.
   • CommentAuthorRodMcGuire
   • CommentTimeSep 3rd 2017
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI've added the tag `endorel` so that one can more easily reference the category $EndoRel$ as [[relation#endorel]]. I've also added the sub sub section [[relation#relation_closures_as_reflexive_subcategories_of_]]. Do relation closures give not just reflective categories but also quasitoposes? E.g. is [[preorder|PreOrd]] a quasitopos? It might take a while if I have to work this out for myself.

   I’ve added the tag endorel so that one can more easily reference the category $EndoRel$ as relation#endorel.

   I’ve also added the sub sub section relation#relation_closures_as_reflexive_subcategories_of_.

   Do relation closures give not just reflective categories but also quasitoposes? E.g. is PreOrd a quasitopos? It might take a while if I have to work this out for myself.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 3rd 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItex[[PreOrd]] is not a quasitopos. (For example, it's not a regular category.) But the category of sets and reflexive relations is a quasitopos, and ditto for symmetric relations and reflexive symmetric relations. I think all these examples are given at [[quasitopos]].

   PreOrd is not a quasitopos. (For example, it’s not a regular category.) But the category of sets and reflexive relations is a quasitopos, and ditto for symmetric relations and reflexive symmetric relations. I think all these examples are given at quasitopos.

6. • CommentRowNumber6.
   • CommentAuthorRodMcGuire
   • CommentTimeSep 6th 2017
   • (edited Sep 6th 2017)
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexin [[relation#relation_closures_as_reflexive_subcategories_of_]] I tweeked and enlarged some wordings, characterized some closure categories and linked to [[quasitopos#exampsep]].

   in relation#relation_closures_as_reflexive_subcategories_of_

   I tweeked and enlarged some wordings, characterized some closure categories and linked to quasitopos#exampsep.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 6th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexBy the way, whenever I explicitly point to a subsection in an entry, I first give it an explicit anchor name. For two reasons: 1. It avoids unpleasant cases of the automatically generated anchor names, as in the present case. 1. (more importantly) it is more robust: Because if at any point in the future anyone decides to give the section an explicit anchor name, then the previous automatically generated anchor name will no longer work. So in the present case I would have first expanded the line #### Relation closures as reflexive subcategories of $EndoRel$ to something like #### Relation closures as reflexive subcategories of $EndoRel$ {#RelationClosuresAsReflexiveSubcategoriesofEndoRel} and then pointed from here to relation#RelationClosuresAsReflexiveSubcategoriesofEndoRel

   By the way, whenever I explicitly point to a subsection in an entry, I first give it an explicit anchor name. For two reasons:

   1. It avoids unpleasant cases of the automatically generated anchor names, as in the present case.

   2. (more importantly) it is more robust: Because if at any point in the future anyone decides to give the section an explicit anchor name, then the previous automatically generated anchor name will no longer work.

   So in the present case I would have first expanded the line

     #### Relation closures as reflexive subcategories of $EndoRel$
   

   to something like

     #### Relation closures as reflexive subcategories of $EndoRel$ {#RelationClosuresAsReflexiveSubcategoriesofEndoRel}
   

   and then pointed from here to

     relation#RelationClosuresAsReflexiveSubcategoriesofEndoRel
   

8. • CommentRowNumber8.
   • CommentAuthorRodMcGuire
   • CommentTimeSep 6th 2017
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexUrs - I do put in explicit tags if I want to link from some page of the nLab. E.g. [[relation#endorel]]. However for directing attention to some section in an nLab discussion, I just rely on the automatically generated `id`s. Editing in a tag and creating a new version just for nForum discussion might be over kill. Most discussions here usually link just the page name. Sure the auto ids might change in the future but that won't cause a big problem in old nForum entries.

   Urs - I do put in explicit tags if I want to link from some page of the nLab. E.g. relation#endorel.

   However for directing attention to some section in an nLab discussion, I just rely on the automatically generated ids. Editing in a tag and creating a new version just for nForum discussion might be over kill. Most discussions here usually link just the page name. Sure the auto ids might change in the future but that won’t cause a big problem in old nForum entries.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeSep 6th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexSure, good. Just thought I'd highlight the issue. Great to hear that you are already doing this!

   Sure, good. Just thought I’d highlight the issue. Great to hear that you are already doing this!

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeSep 6th 2017
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexIt's definitely happened to me that I come back to an old nForum post and want to follow a link that has since broken because of a change.

    It’s definitely happened to me that I come back to an old nForum post and want to follow a link that has since broken because of a change.