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Atrium > Mathematics, Physics & Philosophy: A question on homotopy pullbacks

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  1. In the notations of homotopy pullback, is it true (eventually assuming that X,Y and Z are fibrant) that in case X --> Z is a weak equivalence then also P--> Y is a weak equivalence?
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2009
    • (edited Dec 12th 2009)

    The homotopy pullback of a weak equivalence is a weak equivalence, yes.

    One way to see it is to compute the homotopy pullback in terms of the ordinary pullback over a fibrant replacement of the original diagram: the fibrant replacement diagram has fibrant objects and both morphisms are fibrations. If one of them was a weak equivalence before, it is now an acylic fibration. These are preserved under ordinary pullback, so the pullback morphism P \to Y is then also an acyclic fibration and hence in particular a weak equivalence.

    More generally, homotopy pullbacks (at least in simplicially enriched model categories) model pullback in (oo,1)-catgeories. Their properties are structurally exactly those of ordinary pullbacks, just with everything working up to higher coherent homotopy. But the proof that the pullback in an (oo,1)-category of a morphism that is an equivalence is again an equivalence is entirely parallel to the proof that the ordinary pullback of an isomorphism is an isomorphism.

    • CommentRowNumber3.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 12th 2009
    • (edited Dec 12th 2009)
    Thanks. that was precisely the argument I had in mind, but with the fibrant replacement I ended up with P --> R(Y) <-- Y, where R(Y) is the object in the fibrant replacement diagram over Y, with P -->R(Y) and R(Y) <-- Y both weak equivalences, and this told me that P and Y were weakly equivalent. But in this picture I had been unable to describe the direct arrow P --> Y one has in the universal homotopy cone description of the homotopy pullback. That's why I was wondering whether one needed the additional hypothesis for Y to be a fibrant object (now I clearly see this condition is suffcient to invert -up to homotopy- the acyclic cofibration Y --> R(Y) )
    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeAug 21st 2011

    There is a query at homotopy pullback.The questions answer is ‘Yes’ I think!

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2011
    • (edited Aug 21st 2011)

    It seems you already fixed the entry somehow? Because currently I cannot understand what the question is about. (Or was about.)

    Should we not remove the query box then?

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeAug 22nd 2011

    I fixed the diagram, but left the query so that ’Stephen’ would get the message about the forum. (I hope he has).

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