I’m trying to learn some -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.

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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.

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