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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 16th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI changed "irreflexive graph" to "directed graph" (aka quiver), as such a presheaf is more commonly known. (Irreflexive to my mind means loops at a vertex are *forbidden*.)

   I changed “irreflexive graph” to “directed graph” (aka quiver), as such a presheaf is more commonly known. (Irreflexive to my mind means loops at a vertex are forbidden.)

2. • CommentRowNumber2.
   • CommentAuthorThomas Holder
   • CommentTimeMay 17th 2015
   • (edited May 17th 2015)
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItex'Irreflexive' is admittedly potentially irritating. On the other hand, in the context of the gros-petit distinction where Lawvere uses this terminology it is not unreasonable as 'irreflexive' graphs play second (derived&contrastive) fiddle to reflexive graphs there. He explicitly defends the terminology as 'appropriate' in 'Qualitative distinctions' (1989, p.272). In my view, the nlab should have in principle a consistent terminology for the categories but should also be able to give a meaningful search result for 'irreflexive graphs' be that a redirect link or a footnote with a caveat insofar Lawvere's usage collides with established terminology in graph theory.

   ’Irreflexive’ is admittedly potentially irritating. On the other hand, in the context of the gros-petit distinction where Lawvere uses this terminology it is not unreasonable as ’irreflexive’ graphs play second (derived&contrastive) fiddle to reflexive graphs there. He explicitly defends the terminology as ’appropriate’ in ’Qualitative distinctions’ (1989, p.272).

   In my view, the nlab should have in principle a consistent terminology for the categories but should also be able to give a meaningful search result for ’irreflexive graphs’ be that a redirect link or a footnote with a caveat insofar Lawvere’s usage collides with established terminology in graph theory.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 17th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI put in a parenthetical comment that Lawvere calls them irreflexive graphs, which might suffice for a search result internal to the nLab, but please feel free to do more if you feel it is warranted, so long as "directed graphs" and "quivers" remain. I am not able to access page 272 from Google books, and I was denied access to the article which is hosted by the wordpress blog [Conceptual Mathematics](https://conceptualmathematics.files.wordpress.com/2013/01/toposesofgeneralizedgraphs.pdf) as I am not a member of that blog; I guess it's now by invitation only. So I can't really comment on Lawvere's use of language.

   I put in a parenthetical comment that Lawvere calls them irreflexive graphs, which might suffice for a search result internal to the nLab, but please feel free to do more if you feel it is warranted, so long as “directed graphs” and “quivers” remain.

   I am not able to access page 272 from Google books, and I was denied access to the article which is hosted by the wordpress blog Conceptual Mathematics as I am not a member of that blog; I guess it’s now by invitation only. So I can’t really comment on Lawvere’s use of language.

4. • CommentRowNumber4.
   • CommentAuthorThomas Holder
   • CommentTimeMay 17th 2015
   • (edited May 17th 2015)
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexThat's fine with me. Thanks for enhancing the readability of the entry! I brought the Lawvere paper up mainly because I thought that you might be interested in his point of view. He writes there: Its actions $\mathcal{S}^{\mathbf{P}^{op}}$ are the _irreflexive_ graphs (the negative is in a way appropriate even for those objects which happen to have loops at some point _p_, for morphisms are allowed to interchange _any_ two such loops). (Lawvere 1989, p.272) In a context where one contrasts the petit topos of graphs with the gros it is a natural choice. In a more general situation one might indeed prefer other terms.

   That’s fine with me. Thanks for enhancing the readability of the entry!

   I brought the Lawvere paper up mainly because I thought that you might be interested in his point of view. He writes there:

   Its actions $\mathcal{S}^{\mathbf{P}^{op}}$ are the irreflexive graphs (the negative is in a way appropriate even for those objects which happen to have loops at some point p, for morphisms are allowed to interchange any two such loops). (Lawvere 1989, p.272)

   In a context where one contrasts the petit topos of graphs with the gros it is a natural choice. In a more general situation one might indeed prefer other terms.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeMay 17th 2015
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI'm not convinced that any of these comments argue meaningfully for the *terminology* "irreflexive" as applied to these objects.

   I’m not convinced that any of these comments argue meaningfully for the terminology “irreflexive” as applied to these objects.

6. • CommentRowNumber6.
   • CommentAuthorPeter Heinig
   • CommentTimeJun 17th 2017
   • (edited Jun 17th 2017)
   • PermaLink
   Author: Peter Heinig
   Format: MarkdownItex[[étendue]] had > iff all morphisms in $\mathcal{C}$ are monic. For monoids this amounts to being cancellative: $m_1m=m_2m$ implies $m_1=m_2$ and includes all free monoids. in which the implication appears to have had the order of the factors accidentally wrong (since monoids are not usually assumed commutative by default, and since the fact that $\mathcal{C}^{op}$ is the base category here seems not to make that right; the sentence was about $\mathcal{C}$); simply deleted the implication and made a link to [[left cancellative category]], and slightly reworded.

   étendue had

   iff all morphisms in $\mathcal{C}$ are monic. For monoids this amounts to being cancellative: $m_1m=m_2m$ implies $m_1=m_2$ and includes all free monoids.

   in which the implication appears to have had the order of the factors accidentally wrong (since monoids are not usually assumed commutative by default, and since the fact that $\mathcal{C}^{op}$ is the base category here seems not to make that right; the sentence was about $\mathcal{C}$); simply deleted the implication and made a link to left cancellative category, and slightly reworded.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 17th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSeems like a good call -- thanks.

   Seems like a good call – thanks.

8. • CommentRowNumber8.
   • CommentAuthorThomas Holder
   • CommentTimeOct 22nd 2018
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexI added an argument that slices of étendues are étendues which has the corollary that slices of IAC toposes are IAC toposes as well. It would be good if somebody could check the argument. <a href="https://ncatlab.org/nlab/revision/diff/%C3%A9tendue/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/%C3%A9tendue/21">v21</a>, <a href="https://ncatlab.org/nlab/show/%C3%A9tendue">current</a>

   I added an argument that slices of étendues are étendues which has the corollary that slices of IAC toposes are IAC toposes as well. It would be good if somebody could check the argument.

   diff, v21, current