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1. • CommentRowNumber1.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeSep 2nd 2015
   • PermaLink
   Author: Noam_Zeilberger
   Format: MarkdownItexGiven a functor $U : D \to C$ and an object $x \in C$, the comma categories $U/x$ and $x/U$ can be seen as sort of "$U$-relative" slice and coslice categories: they are equivalently defined as pullbacks along $U$ of the projections from the ordinary slice/coslice categories $C/x$ and $x/C$. I was wondering whether people here know of a commonly accepted name for these specific comma categories, and/or whether they are already discussed somewhere on the nLab. One place where you could imagine this specific concept being useful is when you have some [[concrete category]] $D$ of widgets, and you want to define a new category of "pointed widgets" whose objects are widgets with a distinguished element. Well, in some situations taking the ordinary coslice $1/D$ (as described in the article [[pointed object]]) will not give you what you want, which is really the coslice of 1 relative to the forgetful functor $D \to Set$. Actually, after some googling I see that this is exactly the construction described in [an answer by Todd Trimble to a MO question by Zhen Lin](http://mathoverflow.net/a/50420/1015) about the "category of pointed rings". Has any of this made it into the nLab?

   Given a functor $U : D \to C$ and an object $x \in C$, the comma categories $U/x$ and $x/U$ can be seen as sort of “$U$-relative” slice and coslice categories: they are equivalently defined as pullbacks along $U$ of the projections from the ordinary slice/coslice categories $C/x$ and $x/C$. I was wondering whether people here know of a commonly accepted name for these specific comma categories, and/or whether they are already discussed somewhere on the nLab.

   One place where you could imagine this specific concept being useful is when you have some concrete category $D$ of widgets, and you want to define a new category of “pointed widgets” whose objects are widgets with a distinguished element. Well, in some situations taking the ordinary coslice $1/D$ (as described in the article pointed object) will not give you what you want, which is really the coslice of 1 relative to the forgetful functor $D \to Set$. Actually, after some googling I see that this is exactly the construction described in an answer by Todd Trimble to a MO question by Zhen Lin about the “category of pointed rings”. Has any of this made it into the nLab?

2. • CommentRowNumber2.
   • CommentAuthorThomas Holder
   • CommentTimeSep 2nd 2015
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   Author: Thomas Holder
   Format: MarkdownItexConcerning the terminology, Mac Lane - CWM would call these things 'categories U-over x' , resp. 'U-under x'.

   Concerning the terminology, Mac Lane - CWM would call these things ’categories U-over x’ , resp. ’U-under x’.

3. • CommentRowNumber3.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeSep 2nd 2015
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   Author: Noam_Zeilberger
   Format: MarkdownItexThanks, good to know that Mac Lane had a terminology for this.

   Thanks, good to know that Mac Lane had a terminology for this.

4. • CommentRowNumber4.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeSep 3rd 2015
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   Author: Noam_Zeilberger
   Format: MarkdownItexI'm still interested in whether there is more discussion/use of "$U$-under/over" categories on the nLab. (I see that they are used in [[codense functor]] and [[dense functor]].) Would it be worth adding a short paragraph about these constructions in [[under category]]/[[overcategory]], and another brief section to [[pointed object]] describing situations where one should consider $1/U$ rather than $1/C$?

   I’m still interested in whether there is more discussion/use of “$U$-under/over” categories on the nLab. (I see that they are used in codense functor and dense functor.) Would it be worth adding a short paragraph about these constructions in under category/overcategory, and another brief section to pointed object describing situations where one should consider $1/U$ rather than $1/C$?

5. • CommentRowNumber5.
   • CommentAuthorRichard Williamson
   • CommentTimeSep 3rd 2015
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   Author: Richard Williamson
   Format: MarkdownItexJust a quick note, as nobody else has pointed it out yet: these kind of comma categories are used all the time in abstract homotopy theory, for instance in the formulation of Quillen's 'Theorem A' and 'Theorem B', and (not unrelatedly!) in the theory of test categores/fundamental localisers/derivators. I'm not sure how much there is in the nLab itself on these matters.

   Just a quick note, as nobody else has pointed it out yet: these kind of comma categories are used all the time in abstract homotopy theory, for instance in the formulation of Quillen’s ’Theorem A’ and ’Theorem B’, and (not unrelatedly!) in the theory of test categores/fundamental localisers/derivators. I’m not sure how much there is in the nLab itself on these matters.

6. • CommentRowNumber6.
   • CommentAuthorThomas Holder
   • CommentTimeSep 3rd 2015
   • (edited Sep 3rd 2015)
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexIn CWM the concept plays a role in the proof of the adjoint functor theorem, Mac Lane has even a lemma on it p.121. So no doubt, this is an important concept. On the other hand, the apparent lack of established terminology points to its being rather generally conceived of as an auxiliary construction or special case of a comma category. The concept also plays a role in shape theory where e.g. it appears as comma category of U-objects under x in the Cordier-Porter book (p.28).

   In CWM the concept plays a role in the proof of the adjoint functor theorem, Mac Lane has even a lemma on it p.121. So no doubt, this is an important concept. On the other hand, the apparent lack of established terminology points to its being rather generally conceived of as an auxiliary construction or special case of a comma category.

   The concept also plays a role in shape theory where e.g. it appears as comma category of U-objects under x in the Cordier-Porter book (p.28).

7. • CommentRowNumber7.
   • CommentAuthorRichard Williamson
   • CommentTimeSep 3rd 2015
   • (edited Sep 3rd 2015)
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexRegarding terminology: I would refer to them still simply as slice categories, as the notation ($U / x$ or $D / x$) suggests. There is no possible confusion or conflict with the case where $U$ is an identity functor.

   Regarding terminology: I would refer to them still simply as slice categories, as the notation ($U / x$ or $D / x$) suggests. There is no possible confusion or conflict with the case where $U$ is an identity functor.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 3rd 2015
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   Author: Todd_Trimble
   Format: MarkdownItexRegarding Noam's request for nLab examples and in view of Thomas's #6, we do have a $U$-under in [[adjoint functor theorem]]; there the $U$ is called $R$. We called it there just a comma category.

   Regarding Noam’s request for nLab examples and in view of Thomas’s #6, we do have a $U$-under in adjoint functor theorem; there the $U$ is called $R$. We called it there just a comma category.