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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 κ\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.

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