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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 20th 2009
   • PermaLink
   Author: Urs
   Format: MarkdownI tried at [[locally presentable category]] to incorporate the upshot of the query box discussions into the text, then moved the query boxes to the bottom

   I tried at locally presentable category to incorporate the upshot of the query box discussions into the text, then moved the query boxes to the bottom

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 24th 2009
   • (edited Nov 24th 2009)
   • PermaLink
   Author: Urs
   Format: MarkdownI added to [[small object]] a section <a href="http://ncatlab.org/nlab/show/small+object#details_3">Details</a> with a discussion of how homs out of X commuting with colimits is equivalent to morphisms from X into a colimit having lifts to one of the component objects. Then I added a section <a href="http://ncatlab.org/nlab/show/small+object#properties_4">Properties</a> where I want to deduce from this that hence every small object is <latex> \lambda </latex>-small for _every_ sufficiently large regular cardinal <latex> \lambda </latex> . **Help!** This seems obvious enough, but I feel like I am missing the fully formal way to say this. For instance the fact that <latex> \lambda </latex> be regular must be important, but I am not sure exactly how that goes into the argument. Must be something about colimits over regular-<latex> \lambda </latex>-filtered diagrams being computable as colimits over sub-diagrams of colimits. But I keep feeling unsure about this.

   I added to small object a section Details with a discussion of how homs out of X commuting with colimits is equivalent to morphisms from X into a colimit having lifts to one of the component objects.

   Then I added a section Properties where I want to deduce from this that hence every small object is -small for every sufficiently large regular cardinal .

   Help!

   This seems obvious enough, but I feel like I am missing the fully formal way to say this. For instance the fact that be regular must be important, but I am not sure exactly how that goes into the argument. Must be something about colimits over regular--filtered diagrams being computable as colimits over sub-diagrams of colimits. But I keep feeling unsure about this.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeNov 25th 2009
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownActually, it's simpler than that. I added the proof to [[small object]].

   Actually, it's simpler than that. I added the proof to small object.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2009
   • PermaLink
   Author: Urs
   Format: MarkdownThanks, Mike,now that's indeed obvious enough. I started adding a section listing examples at [[locally small category]]. But this should eventually be extended.

   Thanks, Mike,now that's indeed obvious enough.

   I started adding a section listing examples at locally small category. But this should eventually be extended.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2009
   • PermaLink
   Author: Urs
   Format: Markdownadded to [[accessible category]] a <a href="http://ncatlab.org/nlab/show/accessible+category#relation_to_other_concepts_4">sentence</a> on how that relates to [[locally presentable category]] -- hope I got that right

   added to accessible category a sentence on how that relates to locally presentable category -- hope I got that right

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeNov 25th 2009
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownMinor correction: I think every accessible category is locally small.

   Minor correction: I think every accessible category is locally small.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2009
   • PermaLink
   Author: Urs
   Format: Markdownfleshed out the material at [[locally presentable category]] a bit more - made the distinction between locally presentable and locally kappa-presentable - added more details to the simple example of Set as locally <latex> \aleph_0 </latex>-presentable.

   fleshed out the material at locally presentable category a bit more

   • made the distinction between locally presentable and locally kappa-presentable

   • added more details to the simple example of Set as locally -presentable.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 15th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to the list of equivalent definitions at [[locally presentable category]] the characterization as reflective subcategories of presheaf categories.

   added to the list of equivalent definitions at locally presentable category the characterization as reflective subcategories of presheaf categories.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeApr 15th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThat's only an equivalent *definition* if you also assume the category to be closed under $\kappa$-filtered colimits, although of course what you wrote didn't claim that it was. I modified the statement to make this clear.

   That’s only an equivalent definition if you also assume the category to be closed under $\kappa$-filtered colimits, although of course what you wrote didn’t claim that it was. I modified the statement to make this clear.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexMike, thanks, I may actually be mixed up about this. I guess the point is that I should be talking about _accessible_ localizations where I have so far been talking (and thinking) about just localizations. I'll look into it now...

    Mike, thanks, I may actually be mixed up about this. I guess the point is that I should be talking about accessible localizations where I have so far been talking (and thinking) about just localizations.

    I’ll look into it now…

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexMike, I have edited your edit further now by mentioning that this means that the embedding functor is an [[accessible functor]]. Hope I got it right.

    Mike,

    I have edited your edit further now by mentioning that this means that the embedding functor is an accessible functor. Hope I got it right.