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Stream: theory: mathematics

Topic: Homotopy quotient of B^2U(1)

Alonso Perez-Lona (Aug 28 2025 at 18:01):

Hi all, I am thinking of smooth homotopy quotients but there is something I am not quite clear how to do. For $B^2 U(1)$ the moduli 2-stack of bundle gerbes, one can compute its automorphism 3-group, which can be presented as a 2-crossed module $(U(1) \xrightarrow{0} U(1) \xrightarrow{0} \mathbb{Z}_2)$. I am interested in computing the homotopy quotient under this action. For instance, taking the quotient under $\mathbb{Z}_2$ gives the stack of $\mathbb{Z}_2$-equivariant bundle gerbes. What about that middle $U(1)$, which corresponds to the natural 2-isomorphisms of the identity 2-functor?

Alonso Perez-Lona (Aug 28 2025 at 18:11):

The motivation is the following. For $BG$ the stack of principal $G$ bundles, one can compute its stack of automorphisms $AUT(G)$. This has a very concrete action in terms of bundles/Cech data: for a manifold $M$, given a map $M \to AUT(G)$, which consists of pairs $(\phi_i, g_{ij})$ where $\phi_i: U_i \to Aut(G)$, $g_{ij}: U_{ij} \to G$ satisfying the conditions of a groupoid bundle, one can show that this induces a functor on the category of principal $G$ bundles over $M$ and, furthermore, a gauge transformation induces a natural isomorphism.

So similarly here, I think I should see something similar, namely given a map $M \to AUT(BU(1))$ to the 3-group as a 2-stack, which consists of a $\mathbb{Z}_2$ element on each patch $z_i$ an isomorphism of $U(1)$ , a $U(1)$ -valued function $g_{ij}$ on each double intersection describing a principal $U(1)$ bundle, and another $U(1)$ valued function $h_{ijk}$ on the triple intersection describing a bundle gerbe, these should induce a 2-functor on the 2-category of bundle gerbes, a gauge transformation should induce a natural 2-isomorphism, and a gauge-of-gauge transformation a modification. I think I understand the 2-functors that $z_i$ and $h_{ijk}$ induce but I do not see how the principal $U(1)$ bundle $g_{ij}$ could induce a 2-functor

John Baez (Aug 29 2025 at 11:05):

Does tensoring with a principal U(1) bundle induce a 2-functor on the 2-category of bundle gerbes?

(That may not make sense at all.)

David Michael Roberts (Aug 29 2025 at 12:06):

The groupoid of equivalences between two bundle gerbes, if not empty, is a 2-torsor for the groupoid of U(1)-bundles.

David Michael Roberts (Aug 29 2025 at 12:09):

I'm intrigued by the calculation of the 3-group, though.

David Michael Roberts (Aug 29 2025 at 12:10):

I guess the Z/2 acts by dualising...?