Universal bundles – or mapping cocylinders – 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 groupoidal, comma objects.
Since universal bundles are examples of this construction, we here speak of generalized universal bundles. Another appropriate term might be 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
and in particular
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 generalized universal $B$-bundle is (if it exists) the morphism
which is the total composite vertical morphism of the pullback diagram
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” $g : X \to B$
This can be understood as a “(directed) homotopy pullback” of the point:
If one defines, as one does, a (possiby 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 $g$
The fiber of the generalized universal bundle is the loop monoid $\Omega_{pt} B$:
the sequence
is exact in that $i$ is the kernel of $p$ in the sense of kernels of morphisms of pointed objects (see there).
In (higher) categorical contexts, take the interval object to the the interval category $I := \{a \to b\}$. Then
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
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
(For instance $\hat X$ could be the Čech 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
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)
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.
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
This was shown to be action bigroupoid of $G$ acting on itself in
One can show that every $G$-principal 2-bundle as described in
Toby Bartels, 2-Bundles (arXiv)
Christoph Wockel, …
Igor Bakovič, Bigroupoid 2-torsors PhD thesis, Munich (2008) (pdf).
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
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 toposes:
$\mathbf{E}_{pt} (-1)Cat \to (-1)Cat$ is $\{\top\} \to \{\top, \bottom\}$, the subobject classifier in Set.
$\mathbf{E}_{pt} 0Cat \to 0Cat$ 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.
$\mathbf{E}_{pt} Cat \to Cat$ is $Cat_* \to Cat$. Pullback of this is the Grothendieck construction.
It was David Roberts in the blog comment
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
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 (∞,1)-categories. This is described at universal fibration of (∞,1)-categories.
A morphism $\rho : B \to F$ to a pointed object $F$ (needs not be a basepoint preserving morphism!) can be regarded as a representation of $B$ on the point of $F$. The pullback of the universal $F$-bundle along this morphism
can be addressed as the $F$-bundle $\rho$-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
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: