Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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…