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:

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.