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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    When Lurie says in HTT that an ∞-category is “weakly contractible”, does he mean “categorically equivalent to a point” or “weakly equivalent to a point” (in the sense of the Quillen model structure on SSet)?

    Also, in a general model category, what qualifies as contractible? Obviously “homotopy equivalent to the terminal object” works fine, but is there a nicer condition we can give? Obviously if f:X*f:X\to * where XX is cofibrant and ff is a trivial fibration, it is a homotopy equivalence by the Whitehead theorem, since in that case, X is fibrant-cofibrant, (and hence why in SSet we can simply require that ff be a trivial fibration) (since all objects are cofibrant), but does the converse hold (that is, if X is contractible, then it is fibrant-cofibrant and weakly equivalent to the terminal object)?

    Edit: Nevermind on the first question. Every oo-category is fibrant-cofibrant in the Joyal model structure, so it doesn’t mean categorically equivalent, since then it would follow immediately if that were the definition that every weakly-contractible oo-category is contractible.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2010

    re the second question: a morphism between cofibrant-fibrant objects that is a weak equivalence is also a homotopy equivalence.

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    Yes, of course, this is why I mentioned the Whitehead theorem (this is what it is called in Hirschhorn). I was asking about a converse to that statement in the special case that the target is the terminal object (so we can say X is contractible if and only if it is fibrant-cofibrant, and the unique map X*X\to * is a weak equivalence). It doesn’t seem to be obviously true (and if I had to guess, I would probably guess that it’s not). There’s probably a counterexample, but I can’t think of one offhand.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    The terminal object is necessarily fibrant. Therefore whether it is cofibrant or not, we have for cofibrant XX a weak equivalence of derived hom-space \infty-groupoids

    C(X,*)Hom(X,*). C(X,*) \stackrel{\simeq}{\to} \mathbb{R}Hom(X,*) \,.

    On the right all objects that are equivalences are homotopy equivalences.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2010

    Hm, I guess what I just said doesn’t answer your question…