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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2010
    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 11th 2010

    asked a question at sub-quasi-category.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2010

    thanks, good point. That sounds plausible, given what the pullback diagram there expresses. Let me think about it...

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2010

    I renamed sub-quasi-category to sub-(infinity,1)-category and then edited it a bit, effectively following the suggestion that Mike had made here a while ago in a query box (which is kept at the very end).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2010

    added to sub-(infinity,1)-category and to adjoint (infinity,1)-functor the statements that for (LR)(L \dashv R ) an oo-adjuncton we have

    • RR is full and faithful precisely if the counit is an equivalence

    • LL is full and faithful precisely if the unit is an equivalence.

    the proof is verbatim that for 1-categories, with “verbatim” interpreted in the right way. :-) In HTT I see it only as a paranthetical remark on p. 308.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeApr 29th 2010

    Looks nice, thanks!

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2010

    quick remark on 2-subcategories as 2-subobjects classified by the 2-subobject classifier Set *SetSet_* \to Set. will polish later…

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2010
    • (edited Apr 29th 2010)

    for Lurie’s definition of subcatergory of a quasi-category to be fully equivalent to our notion of 2-sub-(oo,1)-category it would have to be relaxed from the condition hDhCh D \hookrightarrow h C being a subcategory to being any faithful functor, I think. Saying subcategory without meaning either full subcategory of just faithful functor is an evil thing to do anyway.

    We really eventually should write a decent entry n-subobject. We have very good material on this scattered at stuff, structure, property, at generalized universal bundle and a little bit at object classifier. But it would be good to have a more coherent and more comprehensive discussion, eventually.

    I hope it is true for all nn that an (n+1)(n+1)-sub-(,1)(\infty,1)-category of an \infty-groupoid is any \infty-functor DCD \to C that arises, up to equivalence, as an ordinary pullback of nGrpd *nGrpdn Grpd_* \to n Grpd.

    And it is also a pain that we (and Lurie, for that matter, when he talks about his universal Cartesian fibration) talk about ordinary pullbacks here. All these things should instead be lax comma-pullbacks of the point *nGrpd* \to n Grpd.