Is there any treatment on principal "categorical" bundles - principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is some (topological) category?

I know that one can define "categorical covering spaces" - locally constant $\mathbf{C}$-valued sheaves on a topological space $X$. I think that there's a nice equivalence of categories of the form

$$\left\{\text{Locally constant } \mathbf{C}-\text{sheaves on }X \right\} \leftrightarrow [\Pi_1(X), \mathbf{C}]$$

where $\Pi_1(X)$ is the fundamental groupoid of $X$ and $[\_,\_]$ is the functor category.

Is there a general notion of principal $\mathbf{C}-$bundle that captures this particular case? Is there any hope of a nice result in the form of a bijection

$$\left\{ \text{Principal } \mathbf{C}-\text{bundles on } X \text{ up to iso} \right\} \leftrightarrow [X, B \mathbf{C}]$$

where $[X, B\mathbf{C}]$ represents homotopy classes of maps from $X$ to the classifying space of the category $\mathbf{C}$?


2 Answers 2


Such a notion of a principal $\def\bC{{\bf C}}\bC$-bundle (when $\bC$ is a topological or simplicial category, or a Segal space) is available in Definition 6.1 of the paper Classifying spaces of infinity-sheaves.

It works as expected: the cocycle data of a $\bC$-bundle on $X$ is given by an open cover $\{U_i\}_{i∈I}$ of $X$ together with maps $U_i→{\bf Ob}(\bC)$, $U_i∩U_j→{\bf Mor}(\bC)$, etc., which must satisfy the appropriate higher coherence conditions, expressed via the Čech nerve. Some care must be exercised since $B\bC$ inverts all morphisms in $\bC$, which can be taken into account by modifying the Čech nerve appropriately.

Then one can define the space of $\bC$-bundles over $X$ (Definition 6.4) and prove that this space is a representable ∞-sheaf with respect to $X$. When $X$ is a manifold, this is proved in Theorem 6.6 of op. cit. When $X$ is an arbitrary topological space, one can instead cite Theorem 8.5 of Numerable open covers and representability of topological stacks, where open covers used to define $\bC$-bundles must be numerable.

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    $\begingroup$ @მამუკაჯიბლაძე: No, it is not. It is a functor from the modified Čech nerve of $\{U_i\}_{i∈I}$ to $\bf C$. The latter is introduced in Definition 6.1 and has the disjoint union of all finite intersections as objects, and their inclusions as morphisms. $\endgroup$ Mar 26 at 12:45
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    $\begingroup$ Oh wow that's interesting! Thanks. This seems to be a new notion, have not seen it before. So this modified nerve is not equivalent to any kind of groupoid? $\endgroup$ Mar 26 at 12:51
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    $\begingroup$ @მამუკაჯიბლაძე: For example, take an open cover with two elements $\{U,V\}$. The the modified Čech nerve is the category internal to topological spaces whose nonidentity morphisms look like $U←U∩V→V$, and are not invertible. $\endgroup$ Mar 26 at 13:00
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    $\begingroup$ @მამუკაჯიბლაძე: Yes, I see now. The notion described in my answer is essentially the homotopy coherent analogue of the construction described in Johnstone's book. This is also the construction used by Moerdijk in his book “Classifying Spaces and Classifying Topoi”, and also by Weiss in his paper. Roughly, a $\def\bC{{\bf C}}\bC$-bundle over $X$ is an ∞-flat functor $\def\Sh{{\sf Sh}}\bC→\Sh(X)$, where $\Sh(X)$ is the ∞-topos of ∞-sheaves on $X$. To extract such a functor from the above construction, send $A∈\bC$ to the ∞-sheaf given by mapping the Čech cocycle data into $A$. $\endgroup$ Mar 26 at 15:03
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    $\begingroup$ @მამუკაჯიბლაძე: The only source I am aware of is Section 5.3.2 in Lurie's Higher Topos Theory. $\endgroup$ Mar 26 at 15:19

You may find the following paper relevant, by thinking of your topological groupoid as a topological $2$-category with a single object (and invertible $2$-morphisms). The geometric nerve discussed there classifies the corresponding principal $2$-bundles.

Baas, Nils A.; Bökstedt, Marcel; Kro, Tore August
Two-categorical bundles and their classifying spaces.
J. K-Theory 10 (2012), no. 2, 299–369.


It generalizes

Baas, Nils A.; Dundas, Bjørn Ian; Rognes, John
Two-vector bundles and forms of elliptic cohomology.
Topology, geometry and quantum field theory, 18–45,
London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004.


In the presence of a symmetric monoidal structure that you might want to group complete with respect to, the following two papers may be relevant:

Baas, Nils A.; Dundas, Bjørn Ian; Richter, Birgit; Rognes, John
Stable bundles over rig categories.
J. Topol. 4 (2011), no. 3, 623–640.


Baas, Nils A.; Dundas, Bjørn Ian; Richter, Birgit; Rognes, John
Ring completion of rig categories.
J. Reine Angew. Math. 674 (2013), 43–80.


(Apologies for the self-promotion.)

  • $\begingroup$ Dear John, you seem to be one level "too high". A topological 2-category with a single object is a topological 2-group, which is more then a topological groupoid. The question is about ordinary bundles, not about 2-bundles. $\endgroup$ Mar 28 at 6:46

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