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nLab > nLab General Discussions: Remark on Homotopy fiber

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    • CommentRowNumber1.
    • CommentAuthorTintin
    • CommentTimeOct 25th 2015
    Hello:

    I don't know if this is the right place to post this comment. Pardon me if not.

    My remark is the following: At the page "homotopy fiber", when speaking about the "local" definition in category theory it is written for a "category with homotopies" and that some squares "square commutes up to homotopy". I think the author might be willing to say a category with "weak equivalences" and that the squares are in the homotopy category and that they commute "up to weak equivalences".

    I am not an expert, so I might be wrong. Can someone check this? Anyway, thanks.
    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 25th 2015

    What page are you talking about? homotopy fiber redirects to fiber sequence which doesn’t contain the word “local” or the phrases “category with homotopies” or “commutes up to homotopy”.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2015

    Tintin, maybe you mean that page homotopy pullback? That at least speaks about the local and the global definition.

    But in reply to your suggestions:

    I think the author might be willing to say a category with “weak equivalences” and that the squares are in the homotopy category and that they commute “up to weak equivalences”.

    No, weak equivalences are 1-morphisms and a square commutes, if it does, up to a 2-morphism.

    The relation between the two is this: given a 1-category with weak equivalences, then there is the infinity-category (called the “simplicial localization”) obtained by universally turning these weak equivalences into homotopy equivalences. That infinity-category contains 2-morphisms, in general.

    • CommentRowNumber4.
    • CommentAuthorTintin
    • CommentTimeOct 31st 2015
    Sorry for my poorly referred and stated question:

    >Tintin, maybe you mean that page _homotopy pullback_?

    Yes, I meant "Homotopy pullback" page and not "homotopy fiber".

    > No, weak equivalences are 1-morphisms and a square commutes, if it does, up to a 2-morphism

    Yes, it is right that if a square commutes it does up to homotopy, not weak equivalences.

    --------

    However, my suggestion is still up: If I am not mistaken the category needs to have weak equivalences and, on top of that, homotopies.

    The way it is written a newcomer, like me, could think that there might exist categories with "homotopies" without "weak equivalences". Note that when one learns topological spaces, from where notation and intuition is taken, one defines first homotopies and then weak equivalences. Not the other way around.
    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeOct 31st 2015

    If you have homotopies then you also have a natural notion of weak equivalence, namely homotopy equivalence. The most important point for the beginner may be that there are many notions of homotopy pullback which coincide in the most important examples but which could, in principle, be different in general.

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