Question for homotopical readers

I’m trying to learn some \infty-categorical stuff more seriously, and I have a vague question, which maybe a generous reader can offer some insight on. In Lurie’s books, there are a great many different types of “fibration” conditions one can impose on a map of simplicial sets, as in Remark 2.0.0.5 of HTT. The actual definitions aside, how should one “really” think about these different conditions in practice? Basically, I am looking to get some intuition. The only one of these conditions I’ve managed to get a (slight) feeling for is that of a categorical fibration.

One thought on “Question for homotopical readers”

  1. Let me repost a slightly edited version of an e-mail I sent, I hope the readers will gain something from this.

    I distinguish between invariant concepts (i.e. stuff
    that holds in all models of \infty-category theory, for example
    complete Segal anima and quasicategories), and “point-set theory” (stuff
    that comes from the fact that Lurie uses quasi-categories). I
    basically only care about the “image” of point-set stuff under the
    functor sSet–>sSet[Joyal equivalence^-1]=h(\infty-cats), which yields
    invariant notions. (Using the concept “complete Segal anima” already requires to know some version of anima, so naturally it forces you to be invariant, meanwhile the usual complete Segal spaces would have their own point-set concepts, I assume)

    In that sense left/right, (co)cartesian fibrations are invariant concepts
    (corresponding to functors to anima/\infty-categories, sometimes with op
    or not using straightening/unstraightening).
    I should note that (co)cartesian fibrations are sometimes required to be inner fibrations (HTT does this), but inner fibration seem to be a “point-set concept”, more on that later.
    Kan fibrations are those
    left fibrations classifying X–>An such that hX–>hAn factors through
    h(core An). Trivial fibrations are those Kan fibrations with
    contractible fibers, so they correspond to constant functors.
    If you believe in (co)cartesian fibrations as a primitive concepts, then
    left/right fibrations are distinguished between them by saying that the
    fibers are anima or by saying that all edges are (co)cartesian or by
    saying that it is conservative. In that case this might help:
    https://arxiv.org/abs/1510.02402
    Inner fibrations seem to be a point-set concept though, they are just
    functors between \infty-categories (any functor can be factored as inner
    annodyne and inner fibration, inner anodynes are equivalences of
    \infty-categories).

    Maybe you’ll appreciate
    https://www.maths.ed.ac.uk/~cbarwick/papers/fibrations-extra.pdf (which
    also talks about flat fibrations)

    Let me finish by saying that there is a universal left fibration, namely */An–>An. There is also a universal cartesian fibration, which is much harder to describe (it is the (un?)straightening of id: Cat–>Cat formally). Morally it is a lax version of */Cat–>Cat.

    Like

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