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”

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