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\section*{generalized universal bundle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy}{}\paragraph*{{Homotopy}}\label{homotopy}

[[!include homotopy - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{groupoid_incarnations_of_universal_principal_bundles}{Groupoid incarnations of universal principal bundles}\dotfill \pageref*{groupoid_incarnations_of_universal_principal_bundles} \linebreak
\noindent\hyperlink{ordinary_principal_bundles}{Ordinary $G$-principal bundles}\dotfill \pageref*{ordinary_principal_bundles} \linebreak
\noindent\hyperlink{principal_2bundles}{$G$-principal 2-bundles}\dotfill \pageref*{principal_2bundles} \linebreak
\noindent\hyperlink{universal_category_bundles_subobject_classifiers}{Universal $n$-category bundles: $n$-subobject classifiers}\dotfill \pageref*{universal_category_bundles_subobject_classifiers} \linebreak
\noindent\hyperlink{action_groupoids_as_generalized_bundles}{Action groupoids as generalized bundles}\dotfill \pageref*{action_groupoids_as_generalized_bundles} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Universal bundles -- or [[mapping cocylinder]]s -- are intermediate steps in the computation of [[homotopy fibers]], dual to the that way [[mapping cone]] are intermediate steps in the computation of [[homotopy cofibers]].

It is familiar from topology that one can form the path fibration $P X \to X$ of a [[topological space]]. This can be understood as an example of a general construction where one computes [[homotopy pullbacks]] of the [[point]] -- or, if things are not [[groupoid]]al, [[comma objects]].

Since universal bundles are examples of this construction, we here speak of \emph{generalized universal bundles}. Another appropriate term might be \emph{generalized path fibrations}.

One generalizaton of ``generalized universal bundles'' is that the objects in question need not be groupoidal, i.e. they behave like [[directed spaces]]. In this case the [[homotopy pullbacks]] familiar from topology are replaced by [[comma object]] constructions. This is useful in various applications. For instance the constructions [[category of elements]] and [[Grothendieck construction]] can be understood as such directed homotopy pullbacks of the point.

See also

\begin{itemize}%
\item [[fibration sequence]]

\begin{itemize}%
\item [[homotopy limit]]


\item [[homotopy pullback]]



\end{itemize}

\item [[principal bundle]]


\item [[principal 2-bundle]]


\item [[principal ∞-bundle]]



\end{itemize}
and in particular

\begin{itemize}%
\item [[universal principal ∞-bundle]].

\end{itemize}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $C$ be a [[closed monoidal category]] with [[interval object]] $I$. Then for any [[pointed object]] $pt \stackrel{pt_B}{\to}B$ in $C$ the \textbf{generalized universal $B$-bundle} is (if it exists) the morphism

\begin{displaymath}
p : \mathbf{E}_{pt} B \to B
\end{displaymath}
which is the total composite vertical morphism of the [[pullback]] diagram

\begin{displaymath}
\itexarray{
    \mathbf{E}_{\mathrm{pt}}B
    &\to&
    pt
    \\
    \downarrow && \downarrow^{pt_B}
    \\
    [I,B]
    &\stackrel{d_1}{\to}&
    B
    \\
    \downarrow^{d_0}
    \\
    B
  }
  \,.
\end{displaymath}
So the object $\mathbf{E}_{pt} := [I,B]\times_{B} pt$ is defined to be the [[pullback]] of the diagram $[I,B] \stackrel{d_1}{\to} B \stackrel{pt_B}{\leftarrow} pt$ and the morphism $\mathbf{E}_{pt}B \to B$ is the composite of the left vertical morphism in the above diagram which comes from the definition of [[pullback]] and $d_0$.

Then a (generalized) ``$B$-bundle'' on some object $X$ is a morphism $P \to X$ which is the [[pullback]] of the generalized universal $B$-bundle $\mathbf{E}_{pt}$ along a ``classifying morphism'' $g : X \to B$

\begin{displaymath}
\itexarray{
    P &\to& \mathbf{E}_{pt}
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{g}{\to}& B
  }
\end{displaymath}
This can be understood as a ``(directed) [[homotopy pullback]]'' of the point:

If one defines, as one does, a (possiby [[directed object|directed]]) homotopy between two morphisms $f,g : A \to B$ to be a morphism $\eta : A \to [I,B]$ such that $d_0^* \eta = f$ and $d_1^* \eta = g$, then $P$ is the ``lax pullback'' (really [[comma object]]) of the point along $g$

\begin{displaymath}
\itexarray{
    P &\to& *
    \\
    \downarrow &\Downarrow& \downarrow^{pt_B}
    \\
    X &\stackrel{g}{\to}& B
  }
  \,.
\end{displaymath}
The generalized universal bundle can be constructed in this way if we take $X = B$:

\begin{displaymath}
\itexarray{
    \mathbf{E}_{pt} &\to& *
    \\
    \downarrow^{p} &\Downarrow& \downarrow^{pt_B}
    \\
    B &\stackrel{id}{\to}& B
  }
  \,.
\end{displaymath}
The [[fiber]] of the generalized universal bundle is the [[loop space object|loop monoid]] $\Omega_{pt} B$:

\begin{displaymath}
\itexarray{
    \Omega_{pt} B &\to& \mathbf{E}_{pt} &\to& *
    \\
    \downarrow & & \downarrow^{p} &\Downarrow& \downarrow^{pt_B}
    \\
    * &\stackrel{pt_B}{\to} & B &\stackrel{id}{\to}& B
  }
  \,.
\end{displaymath}
the sequence

\begin{displaymath}
\Omega_{pt}B \stackrel{i}{\to} \mathbf{E}_{pt} B \stackrel{p}{\to} B
\end{displaymath}
is exact in that $i$ is the kernel of $p$ in the sense of kernels of morphisms of [[pointed objects]] (see there).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{groupoid_incarnations_of_universal_principal_bundles}{}\subsubsection*{{Groupoid incarnations of universal principal bundles}}\label{groupoid_incarnations_of_universal_principal_bundles}

In (higher) categorical contexts, take the interval object to the the interval category $I := \{a \to b\}$. Then

\hypertarget{ordinary_principal_bundles}{}\paragraph*{{Ordinary $G$-principal bundles}}\label{ordinary_principal_bundles}

For $C =$ [[Cat]], $B := \mathbf{B}G$ a one-object groupoid corresponding to a group $G$ with the unique point, $\mathbf{E}_{pt} \mathbf{B}G = \mathbf{E}G = G//G$ is the [[action groupoid]] of $G$ acting on itself. The sequence of groupoids is

\begin{displaymath}
G \to G//G \to \mathbf{B}G
  \,.
\end{displaymath}
This is the universal $G$-bundle in its groupoid incarnation. It is a theorem by Segal from the 1960s that indeed this maps, under [[geometric realization]] to the familiar universal $G$-bundle in $Top$. Moreover, it can be seen that every $G$-principal bundle $P \to X$ in the ordinary sense is the pullback of $\mathbf{E} G$ in the following sense:

the $G$-bundle $P \to X$ is classified by a nonabelian $G$-valued 1-cocycle (the transition function of any of its local trivializations), which is an [[anafunctor]]

\begin{displaymath}
\itexarray{
    \hat X &\stackrel{g}{\to}& \mathbf{B}G
    \\
    \downarrow^\pi
    \\
    X
  }
  \,.
\end{displaymath}
(For instance $\hat X$ could be the ech groupoid of a [[cover]] of $X$.)

The universal groupoid bundle $\mathbf{E}G \to \mathbf{B}G$ may now be pulled back along this anafunctor to yield the groupoid bundle $g^* \mathbf{E}G \to X$ given by the total left vertical morphism in

\begin{displaymath}
\itexarray{
    g^* \mathbf{E}G &\to& \mathbf{E}G
    \\
    \downarrow && \downarrow
    \\
    \hat X &\stackrel{g}{\to}& \mathbf{B}G
    \\
    \downarrow^\pi
    \\
    X
  }
  \,.
\end{displaymath}
This bundle of groupoids is weakly equivalent to the $G$-principal bundle we started with in that there is a morphism of bundles of groupoids (with $P$ regarded as a bundle of discrete groupoids)

\begin{displaymath}
\itexarray{
     g^* \mathbf{E}G &&\stackrel{\simeq}{\to}&& P
     \\
     & \searrow && \swarrow
     \\  
     &&X
  }
  \,.
\end{displaymath}
In fact that horizontal morphism is an acyclic fibration in the [[folk model structure]], i.e. a [[k-surjective functor]] for all $k$.

This is recalled in the following reference.

\hypertarget{principal_2bundles}{}\paragraph*{{$G$-principal 2-bundles}}\label{principal_2bundles}

For $C = 2Cat$, strict 2-categories , $B := \mathbf{B}G$ a strict one-object 2-groupoid corresponding to a strict [[2-group]] $G$ with the unique point, $\mathbf{E}_{pt} \mathbf{B}G = \mathbf{E}G$ was described under the name $INN(G)$ in

\begin{itemize}%
\item Urs Schreiber, David Roberts, \emph{The inner automorphism 3-group of a strict 2-group}, Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 193-244 (\href{http://arxiv.org/abs/0708.1741v2}{arXiv})

\end{itemize}
This was shown to be \emph{action bigroupoid} of $G$ acting on itself in

\begin{itemize}%
\item Igor Bakovi, \emph{Bigroupoid 2-torsors} PhD thesis, Munich (2008) (\href{http://edoc.ub.uni-muenchen.de/9209/}{pdf}).

\end{itemize}
One can show that every $G$-principal 2-bundle as described in

\begin{itemize}%
\item Toby Bartels, \emph{2-Bundles} (\href{http://arxiv.org/abs/math.CT/0410328}{arXiv})


\item Christoph Wockel, \emph{\ldots{}}


\item Igor Bakovi, \emph{Bigroupoid 2-torsors} PhD thesis, Munich (2008) (\href{http://edoc.ub.uni-muenchen.de/9209/}{pdf}).



\end{itemize}
indeed is recovered as the pullback of $\mathbf{E} G \to \mathbf{B}G$ along the corresponding cocycle, along the lines described above.

The way this works is indicated briefly in the last section of Roberts-Schreiber above. A more detailed description for the moment is in the notes

\begin{itemize}%
\item Urs Schreiber: [[2bundrecon.pdf:file]]

\end{itemize}
\hypertarget{universal_category_bundles_subobject_classifiers}{}\subsubsection*{{Universal $n$-category bundles: $n$-subobject classifiers}}\label{universal_category_bundles_subobject_classifiers}

One can take $B$ to be something very different from the familiar classifying groupoids. Taking it to be $n Cat$ yields the [[subobject classifiers]] of higher [[topos|toposes]]:

\begin{itemize}%
\item $\mathbf{E}_{pt} (-1)Cat \to (-1)Cat$ is $\{\top\} \to \{\top, \bottom\}$, the [[subobject classifier]] in [[Set]].


\item $\mathbf{E}_{pt} 0Cat \to 0Cat$ is $Set_* \to Set$, the forgetful functor from [[pointed sets]], which is the 2-[[subobject classifier]] in [[Cat]]. Pullback of this creates the [[category of elements]] of a [[presheaf]].


\item $\mathbf{E}_{pt} Cat \to Cat$ is $Cat_* \to Cat$. Pullback of this is the [[Grothendieck construction]].



\end{itemize}
It was David Roberts in the blog comment

\begin{itemize}%
\item David Roberts, \href{http://golem.ph.utexas.edu/category/2008/01/101_things_to_do_with_a_2class.html#c014559}{comment in: 101 things to do with a 2-classifier}

\end{itemize}
who first pointed out that these (higher) subobject classifiers are just generalized universal bundles in the above sense.

These cases for $n= 0$ and $n=1$ have been considered in the context of universal category bundles in

\begin{itemize}%
\item Kathryn Hess, \emph{[[HessLackBundCat.pdf:file]]} .

\end{itemize}
The discussion there becomes more manifestly one of bundles if one regards all morphisms $C \to Set$ appearing there as being the right legs of [[anafunctors]].

There is a well-understood version of this for $n = (\infty,1)$, i.e. for [[(infinity,1)-category|(∞,1)-categories]]. This is described at [[universal fibration of (∞,1)-categories]].

\hypertarget{action_groupoids_as_generalized_bundles}{}\subsubsection*{{Action groupoids as generalized bundles}}\label{action_groupoids_as_generalized_bundles}

A morphism $\rho : B \to F$ to a [[pointed object]] $F$ (needs not be a basepoint preserving morphism!) can be regarded as a \emph{representation} of $B$ on the point of $F$. The pullback of the universal $F$-bundle along this morphism

\begin{displaymath}
\rho^* \mathbf{E}_{pt} F \to B
\end{displaymath}
can be addressed as the \emph{$F$-bundle \textbf{$\rho$-associated} to the universal $B$-bundle $\mathbf{E}_{pt}B$.}

If $B$ is a groupoid, then $\rho^* \mathbf{E}_{pt} F$ is the [[action groupoid]] of $B$ acting on the point of $F$.

Further pulling this back along a cocycle $g : \hat X \to B$ of a $B$-principal bundle yields the $\rho$-accociated bundle of that.

For instance for $B = \mathbf{B}G$ and $F = Vect$ with $\rho : \mathbf{B}G \to Vect$ a [[representation]] of the group $G$ on a vector space $V$, the $\rho$-associated $\mathrm{Vect}$-bundle on $\mathbf{B}G$ is

\begin{displaymath}
V \to V//G \to \mathbf{B}G
  \,.
\end{displaymath}
Pulling that further back along the cocycle $g : \hat X \to \mathbf{B}G$ classifying a $G$-principal bundle $P \to X$, one obtains the familiar vector bundle $P \times_G V \to X$ which is $\rho$-associated to $P$, along the lines described above:

\begin{displaymath}
\itexarray{
     g^* \rho^*\mathbf{E}_{pt}F 
    &&\stackrel{\simeq}{\to}&& P\times_G V
     \\
     & \searrow && \swarrow
     \\  
     &&X
  }
  \,.
\end{displaymath}
[[!redirects generalized universal bundles]]



\end{document}