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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 11th 2010
   • PermaLink
   Author: Urs
   Format: Markdowncreated [[sub-quasi-category]]

   created sub-quasi-category

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 11th 2010
   • PermaLink
   Author: Mike Shulman
   Format: Markdownasked a question at [[sub-quasi-category]].

   asked a question at sub-quasi-category.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 12th 2010
   • PermaLink
   Author: Urs
   Format: Markdownthanks, good point. That sounds plausible, given what the pullback diagram there expresses. Let me think about it...

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 28th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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).

   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).

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 29th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to [[sub-(infinity,1)-category]] and to [[adjoint (infinity,1)-functor]] the statements that for $(L \dashv R )$ an oo-adjuncton we have * $R$ is full and faithful precisely if the counit is an equivalence * $L$ 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.

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

   • $R$ is full and faithful precisely if the counit is an equivalence

   • $L$ 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.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeApr 29th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexLooks nice, thanks!

   Looks nice, thanks!

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 29th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexquick remark on 2-subcategories as <a href="http://ncatlab.org/nlab/show/sub-(infinity%2C1)-category#2Subobjects">2-subobjects</a> classified by the 2-subobject classifier $Set_* \to Set$. will polish later...

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

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 29th 2010
   • (edited Apr 29th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor 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 $h D \hookrightarrow h C$ 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 $n$ that an $(n+1)$-sub-$(\infty,1)$-category of an $\infty$-groupoid is any $\infty$-functor $D \to C$ that arises, up to equivalence, as an ordinary pullback of $n Grpd_* \to n Grpd$. 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 $* \to n Grpd$.

   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 $h D \hookrightarrow h C$ 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 $n$ that an $(n+1)$-sub-$(\infty,1)$-category of an $\infty$-groupoid is any $\infty$-functor $D \to C$ that arises, up to equivalence, as an ordinary pullback of $n Grpd_* \to n Grpd$.

   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 $* \to n Grpd$.