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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeApr 8th 2014
   • (edited Apr 9th 2014)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThere is a dead link at [[ionad]] (the one to `web'. This is repeated at [[Richard Garner]]. There is a second link, (to an ArXiv version), so I will delete the dead ones. Is there a published version somewhere as well?

   There is a dead link at ionad (the one to ‘web’. This is repeated at Richard Garner. There is a second link, (to an ArXiv version), so I will delete the dead ones. Is there a published version somewhere as well?

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 9th 2014
   • (edited Apr 9th 2014)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI added the published version and a link, which is >Garner, Richard, _Ionads_. J. Pure Appl. Algebra 216 (2012), no. 8-9, 1734–1747. And the self-archived version is [here](http://comp.mq.edu.au/~rgarner/Papers/Ionads.pdf)

   I added the published version and a link, which is

   Garner, Richard, Ionads. J. Pure Appl. Algebra 216 (2012), no. 8-9, 1734–1747.

   And the self-archived version is here

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 9th 2014
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexEdited [[Richard Garner]] as well as [[ionad]].

   Edited Richard Garner as well as ionad.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 22nd 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexMike Haskel added commentary in the third query box here, which is not a good way to be noticed. In any case, we looking to remove these boxes.

   Mike Haskel added commentary in the third query box here, which is not a good way to be noticed. In any case, we looking to remove these boxes.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeAug 22nd 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexQuery boxes incorporated and removed. Here's the last one, which is probably worth preserving: > It seems to me that a full categorification would allow $X$ to be a [[groupoid]]. ---Toby > [[Mike Shulman]]: Actually, if you want to allow groupoids, I don't think there's any reason not to allow arbitrary categories. Richard and I had a discussion about this question, and at one point I think he was on the side you present, but we've both since come around to this version. Notice that any ionad *induces* a category structure on its set of points, since each point is in particular a geometric morphism from $Set$ to $\Omega(X)$. I think this induced "category of points" should be regarded as a categorification of the [[specialization order]] *induced* on the points of a topological space. In particular, it comes for free as part of the structure; you don't have to specify it in advance. > You *can* specify either of them in advance; you can start with $X$ being a category in the definition of ionad, or you can define a generalized sort of topological space as a poset equipped with a lex comonad on its poset of downsets. In either case it amounts to specifying an ordinary ionad/space, together with a distinguished category/poset mapping bijectively-on-objects to its induced category/poset of points. In both cases, any continuous map necessarily preserves the induced category/order, but if you start with a distinguished category/order of points, your continuous maps have to preserve that too. These notions might be interesting, but the comparison makes me fairly sure that ionads starting with a *set* are already the natural "fully categorified" categorification of topological space. > _Toby_: I didn\'t want to allow just any category, to be analogous to not allowing just any p(r)oset as points of a topological space. But I\'ll think about what happens when the points are allowed to form a groupoid. > [[Mike Shulman]]: I think the same argument applies to groupoids. As I said, any ionad induces a category structure on its set of points, and if you start with a groupoid you're just picking a distinguished groupoid on that set of points which maps bijectively-on-objects to the induced category. > It's like the difference between a [[Segal space]] and a [[complete Segal space]]. Segal spaces may be interesting, but it's the complete ones that model $(\infty,n)$-categories. And also the difference between an ordinary multicategory and an "enhanced" multicategory in the sense of Baez-Dolan-Cheng. The weird notions that come with an extra groupoid structure on the objects may be technically useful, but usually the more fundamental notion is the one where the groupoid of objects is induced from the category of objects. > _Mike Haskel_: I see Mike S.'s point, but I don't think the approach taken in this article is the best resolution: it seems to generalize strangely if you try to define infinity ionads, and it creates some size issues. Thinking about the analogy to complete Segal spaces, I'd rather allow the points to form a category to begin with and require some property of the induced functor from the original points to the points of the topos of opens. (Maybe full and faithful? Requiring it to be part of an equivalence seems too strong, like requiring topological space to be sober.) > Regarding generalizing strangely to "infinity ionad," I was thinking about what happens when you try to find the set of points of an infinity topos. The natural structure on the points is an infinity category. You would then take the [[core]], yielding an infinity groupoid. The strategy in this article would suggest that we should then take the set of connected components, but that seems strange. What's so special about the 0-truncation? And if we want to take the core but don't want to truncate, then that would suggest Toby's original approach in the case of ordinary toposes/ionads. > I also think my suggested approach could possibly clean up the treatment of some size issues. (I'm not really an expert on size issues, so take this with a grain of salt.) Given a topos, the category of points may be large, but it will at least be accessible (I think?). The set of isomorphism classes, however, may be a large set. In order for the construction of an ionad associated with a topos to be canonical, then, we need to allow the set of points $X$ to be a large set. But then $\operatorname{Set}^X$ is nasty, too large to be even a large category. If $X$ is an accessible category, however, then $\operatorname{Set}^X$ should be something like the category of accessible functors, which shouldn't be so bad. (This analysis also suggests that we should allow the original points to be a category rather than a mere groupoid, since the core of the category of points of a topos need not be accessible in any nice way.)

   Query boxes incorporated and removed. Here’s the last one, which is probably worth preserving:

   It seems to me that a full categorification would allow $X$ to be a groupoid. —Toby

   Mike Shulman: Actually, if you want to allow groupoids, I don’t think there’s any reason not to allow arbitrary categories. Richard and I had a discussion about this question, and at one point I think he was on the side you present, but we’ve both since come around to this version. Notice that any ionad induces a category structure on its set of points, since each point is in particular a geometric morphism from $Set$ to $\Omega(X)$. I think this induced “category of points” should be regarded as a categorification of the specialization order induced on the points of a topological space. In particular, it comes for free as part of the structure; you don’t have to specify it in advance.

   You can specify either of them in advance; you can start with $X$ being a category in the definition of ionad, or you can define a generalized sort of topological space as a poset equipped with a lex comonad on its poset of downsets. In either case it amounts to specifying an ordinary ionad/space, together with a distinguished category/poset mapping bijectively-on-objects to its induced category/poset of points. In both cases, any continuous map necessarily preserves the induced category/order, but if you start with a distinguished category/order of points, your continuous maps have to preserve that too. These notions might be interesting, but the comparison makes me fairly sure that ionads starting with a set are already the natural “fully categorified” categorification of topological space.

   Toby: I didn't want to allow just any category, to be analogous to not allowing just any p(r)oset as points of a topological space. But I'll think about what happens when the points are allowed to form a groupoid.

   Mike Shulman: I think the same argument applies to groupoids. As I said, any ionad induces a category structure on its set of points, and if you start with a groupoid you’re just picking a distinguished groupoid on that set of points which maps bijectively-on-objects to the induced category.

   It’s like the difference between a Segal space and a complete Segal space. Segal spaces may be interesting, but it’s the complete ones that model $(\infty,n)$-categories. And also the difference between an ordinary multicategory and an “enhanced” multicategory in the sense of Baez-Dolan-Cheng. The weird notions that come with an extra groupoid structure on the objects may be technically useful, but usually the more fundamental notion is the one where the groupoid of objects is induced from the category of objects.

   Mike Haskel: I see Mike S.’s point, but I don’t think the approach taken in this article is the best resolution: it seems to generalize strangely if you try to define infinity ionads, and it creates some size issues. Thinking about the analogy to complete Segal spaces, I’d rather allow the points to form a category to begin with and require some property of the induced functor from the original points to the points of the topos of opens. (Maybe full and faithful? Requiring it to be part of an equivalence seems too strong, like requiring topological space to be sober.)

   Regarding generalizing strangely to “infinity ionad,” I was thinking about what happens when you try to find the set of points of an infinity topos. The natural structure on the points is an infinity category. You would then take the core, yielding an infinity groupoid. The strategy in this article would suggest that we should then take the set of connected components, but that seems strange. What’s so special about the 0-truncation? And if we want to take the core but don’t want to truncate, then that would suggest Toby’s original approach in the case of ordinary toposes/ionads.

   I also think my suggested approach could possibly clean up the treatment of some size issues. (I’m not really an expert on size issues, so take this with a grain of salt.) Given a topos, the category of points may be large, but it will at least be accessible (I think?). The set of isomorphism classes, however, may be a large set. In order for the construction of an ionad associated with a topos to be canonical, then, we need to allow the set of points $X$ to be a large set. But then $\operatorname{Set}^X$ is nasty, too large to be even a large category. If $X$ is an accessible category, however, then $\operatorname{Set}^X$ should be something like the category of accessible functors, which shouldn’t be so bad. (This analysis also suggests that we should allow the original points to be a category rather than a mere groupoid, since the core of the category of points of a topos need not be accessible in any nice way.)

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeAug 22nd 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexNow let me respond to Mike Haskel. First, I think you're right that another approach would be to allow the points to be a groupoid or a category and then impose a "completeness" condition. I've added that possibility to the page. Thanks! I don't think a categorification would deal with the 0-truncation, though; instead it would be talking about a set that surjects onto the core. I haven't thought about whether there is a way to (generalize the notion of ionad so that we can) make an arbitrary topos into an ionad, analogous to the "space of points" of a locale. It's possible that accessibility would help, but I don't really have an intuition for whether it would be good enough. It might be worth asking Richard; he might have thought about that question.

   Now let me respond to Mike Haskel. First, I think you’re right that another approach would be to allow the points to be a groupoid or a category and then impose a “completeness” condition. I’ve added that possibility to the page. Thanks!

   I don’t think a categorification would deal with the 0-truncation, though; instead it would be talking about a set that surjects onto the core.

   I haven’t thought about whether there is a way to (generalize the notion of ionad so that we can) make an arbitrary topos into an ionad, analogous to the “space of points” of a locale. It’s possible that accessibility would help, but I don’t really have an intuition for whether it would be good enough. It might be worth asking Richard; he might have thought about that question.