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created sub-quasi-category
asked a question at sub-quasi-category.
thanks, good point. That sounds plausible, given what the pullback diagram there expresses. Let me think about it...
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).
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.
Looks nice, thanks!
quick remark on 2-subcategories as 2-subobjects classified by the 2-subobject classifier $Set_* \to Set$. will polish later…
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 $* \to n Grpd$.
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