\[\newcommand{\rv}[1]{\mathbf{#1}} \newcommand{\x}{\rv x} \newcommand{\y}{\rv y} \newcommand{\bar}[1]{\overline{#1}} \newcommand{\wtil}[1]{\widetilde{#1}} \newcommand{\what}[1]{\widehat{#1}} \newcommand{\ep}{\varepsilon} \newcommand{\ph}{\varphi} \newcommand{\maps}{\colon} \newcommand{\to}{\rightarrow} \newcommand{\xraw}{\xrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\To}{\Rightarrow} \renewcommand{\dot}{\centerdot} \renewcommand{\tensor}{\otimes} \newcommand{\pr}{^\prime} \newcommand{\op}[1]{#1^{\mathrm{op}}} \newcommand{\hom}{\textit{Hom}} \newcommand{\oo}{\circ} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\str}[1]{#1^{*}} \newcommand{\intvl}{\mathbb{I}} \newcommand{\yon}{\mathcal{Y}} \newcommand{\topos}{\mathfrak{Top}} \newcommand{\bTopos}{\mathfrak{BTop}} \newcommand{\BTop}{\mathfrak{BTop}} \newcommand{\con}{\mathfrak{Con}} \newcommand{\pos}{\mathfrak{Poset}} \newcommand{\topl}{\mathbb{Top}} \newcommand{\grpd}{\mathbb{Grpd}} \newcommand{\ab}{\mathbf{Ab}} \newcommand{\thT}{\mathbb{T}} \newcommand{\mod}{\mathbf{Mod}} \newcommand{\th}{\mathbf{Th}} \newcommand{\cl}{\mathbf{Cl}} \newcommand{\ob}{\mathbf{Ob}} \newcommand{\aut}{\mathbf{Aut}} \newcommand{\bun}{\mathbf{Bun}} \newcommand{\geom}{\mathbf{Geom}} \def\catg{\mathop{\mathcal{C}\! {\it at}}\nolimits} \def\Cat{\mathop{\mathfrak{Cat}}} \def\Con{\mathop{\mathfrak{Con}}} \def\CAT{\mathop{\mathbf{2} \mathfrak{Cat}}} \def\Cat{\mathop{\mathfrak{Cat}}} \newcommand{\Topos}{\mathfrak{Top}} \newcommand{\ETopos}{\mathcal{E}\mathfrak{Top}} \newcommand{\BTopos}{\mathcal{B}\mathfrak{Top}} \newcommand{\GTopos}{\mathcal{G}\mathfrak{Top}} \newcommand{\Psh}{\textit{Psh}} \newcommand{\Sh}{\textit{Sh}} \newcommand{\psh}[1]{\textit{Psh}(\cat{#1})} \newcommand{\sh}[1]{\textit{Sh}(\cat{#1})} \newcommand{\Id}{\operatorname{Id}} \newcommand{\Ho}{\operatorname{Ho}} \newcommand{\ad}{\operatorname{ad}} \newcommand{\Adj}{\operatorname{Adj}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Set}{\operatorname{Set}} \newcommand{\Pull}{\operatorname{Pull}} \newcommand{\Push}{\operatorname{Push}} \newcommand{\dom}{\operatorname{dom}} \newcommand{\cod}{\operatorname{cod}} \newcommand{\fun}{\operatorname{Fun}} \newcommand{\colim}{\operatorname{Colim}} \newcommand{\cat}[1]{\mathcal{#1}} \newcommand{\scr}[1]{\mathscr{#1}} \newcommand{\frk}[1]{\mathfrak{#1}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\CS}{\mathcal{S}} \newcommand{\CE}{\mathcal{E}} \newcommand{\CF}{\mathcal{F}}\]Principal bundles vs higher principal bundles
The notion of principal bundle is by now a very important classical notion in mathematics, particularly in topology and differential geometry as well as in physics as foundational framework of gauge theories. Perhaps its first robust appearance is in
STEENROD, N. (1951). The Topology of Fibre Bundles. (PMS-14). PRINCETON, NEW JERSEY: Princeton University Press.
First we have to make clear what we mean by the word “higher”. Fortunately for us, there are prior work of J.Lurie in Higher Topos Theory
and also work of Thomas Nikolaus, Urs Schreiber, Danny Stevenson in Principal ∞-bundles – General theory
which gives a theory of higher topos theory and higher principal bundles. From the latter we learn that objects of an $\infty$-topos as geometric homotopy types.
In as far as an ordinary topos is a context for general geometry, an $\infty$-topos is a context for what is called higher geometry or derived geometry: the pairing of the notion of geometry with that of homotopy. (Here derived alludes to derived category and derived functor in homological algebra, but refers in fact to a nonabelian generalization of these concepts.) Therefore we may refer to objects of an $\infty$-topos also as geometric homotopy types.
In these notes we will try to explain work of T.Nikolaus, et al by comparing it to the situation of 1-toposes (aka toposes).
