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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
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.
Actually, it's simpler than that. I added the proof to small object.
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.
added to accessible category a sentence on how that relates to locally presentable category -- hope I got that right
Minor correction: I think every accessible category is locally small.
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.
added to the list of equivalent definitions at locally presentable category the characterization as reflective subcategories of presheaf categories.
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.
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…
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.
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