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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 $f:X\to *$ where $X$ is cofibrant and $f$ 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 $f$ 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 $X\to *$ 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 $X$ a weak equivalence of derived hom-space $\infty$-groupoids

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

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…