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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 20th 2009

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 24th 2009
• (edited Nov 24th 2009)

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 $\lambda$-small for every sufficiently large regular cardinal $\lambda$ .

Help!

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

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2009

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

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 25th 2009

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 25th 2009

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

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2009

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

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 26th 2009

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 $\aleph_0$-presentable.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 15th 2010

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

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeApr 15th 2010

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.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 15th 2010

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…

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 15th 2010

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.