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It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]_{\text{Cat}} \cong \text{Cov}(X)$ where $\text{Cov}(X)$ is the category of covering spaces of $X$, and this equivalence is natural in $X$.

Moreover, $\text{Top}$ classifies fibrations among $(\infty, 1)$-categories. That is, for each topological space $X$, there is an equivalence of $(\infty, 1)$-categories $[ \Pi_{\infty} (X), \text{Top} ]_{(\infty, 1) \text{-Cat} } \cong \text{Fib}(X)$, where $\text{Fib}(X)$ is the category of fibrations over $X$ (this is natural in $X$, too). This is what I was reading in the excerpt below:

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This is quite nice because it seems like fibrations fit into the context of descent, so that we might presume to have the Grothendieck construction and all that.

My questions are:

1) Does this give a sort of $(\infty, 1)$-descent along fibrations, just like ordinary descent is for a Grothendieck fibration, but for $(\infty, 1)$-categories?

2) Is there some dual situation for cofibrations? Is there some $(\infty, 1)$-category $C$ such that $[C, \Pi_{\infty}(X)]_{(\infty, 1) \text{-cat}} \cong \text{Cof}(X)$? Is there a codescent?

3) Does this give an analogous theory for $\Pi_i(X), i \in \mathbb{N}_{\geq 1}$ to the one that exists for $\Pi_1(X)$? I'm not quite sure how to analogize here.

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    $\begingroup$ First, you mean [Π_∞(X), Top] = Fib(X). I'm not sure what precise statement you want when you say descent/codescent. Note that F -> E -> B being a fiber sequence is precisely the claim that E is the colimit of F over B (this is exactly the Grothendieck construction). Next, a map A -> X of spaces is a cofibration if and only if the map Y^X -> Y^A is a fibration for every space Y. You could maybe use this fact to classify cofibrations. As for qn 3, you could restrict attention to homotopy i-types X; then, Π_i(X) = Π_∞(X) in your notation. $\endgroup$
    – skd
    Commented Jul 17, 2019 at 2:39

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  1. Yes. In roughly your language, the forgetful $(\infty,1)$-functor $\rm Fib\to Top$ is an $\infty$-fibration, where the fiber over a space $X$ is the category of fibrations over $X$, and descent in this $\infty$-fibration corresponds to the fact that there is a fibration classifier. This is described in $\infty$-categorical language in section 6.1 of Lurie's Higher topos theory, and in model-categorical language in Rezk's "Toposes and homotopy toposes". Note, though, that once you are talking about $(\infty,1)$-categories there is no difference between fibrations and arbitrary maps, so the above "$\rm Fib$" is actually just the arrow $(\infty,1)$-category of $\rm Top$.

  2. Not as far as I know.

  3. Yes. The analogue of $\rm Set$ when $n=0$ and $\rm Top$ when $n=\infty$ is $n \rm Typ$, the space of homotopy $n$-types.

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