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.

]]>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…

]]>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.

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

]]>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.

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

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

]]>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.

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

]]>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.

]]>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

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