If G is a (discrete 1-)group, then any G-set decomposes uniquely as a coproduct of transitive G-sets, each of which is the quotient of G by a subgroup. Is something analogous true for actions of 2-groups on groupoids, or for n-groups on (n-1)-groupoids? I remember there was a lot of Cafe discussion about higher Klein geometry, subgroups and homogeneous spaces of higher groups, but I don’t remember whether something like this came up.

]]>I’m not sure where to ask this. I thought the n-forum might have the right audience. I’m not sure if this is how the n-forum is supposed to be used, and I apologize if this question is inappropriate. Anyway I’m looking for a reference for some version of the following claim:

Claim: The forgetful functor from 2-groups to categories has a (weak) left adjoint and the corresponding weak adjunction is monadic, by which I mean there is a 2-monad on categories whose algebras are precisely the 2-groups.

It seems pretty clear to me that this should be the case and I am sure this has been thought about extensively already. I just don’t know where to look. I need references. Thanks!

]]>I should have known, but I’ve actually had an intuition of $\mathbf{B}^n U(1)$ as an $(n+1)$-group only a few minutes ago. I’ll share here this point of view for the two or three readers which should happen to be unaware of it (I’m sure this is widely known).

The starting point is the Lie group $U(1)$.

Next, we consider $\mathcal{B}U(1)$, the classifying space for principal $U(1)$-bundles, or equivalently of complex line bundles. And let us look to line bundles not up to isomorphism, but as a category. Tensor product and dual of line bundles make line bundles a group-like category, i.e., a 2-group. Moreover, since tensor product is symmetric, this is an abelian 2-group. More formally, we should think of $\mathbf{B}U(1)$ as the functor mapping a toplogical space $X$ to the 2-group of principal $U(1)$-bundles over $X$.

So far we have gone from the Lie group $U(1)$ to the 2-group $\mathbf{B}U(1)$ of principal $U(1)$-bundles. Next step is going to (the 3-group of) principal $\mathbf{B}U(1)$-bundles. Therefore, on each open set $U_i$ of an open cover of a topological space $X$ we’ll have the 2-group of line bundles on $U_i$. Transition functions will be given by line bundles on $U_{ij}$, acting by tensor product. On triple intersections we’ll have the tensor product of three line bundles which has to be trivialised in a coherent way. This should define a $\mathbf{B}U(1)$-gerbe, and so we are led to think of $\mathbf{B}^2U(1)$ as the functor mapping a topological space $X$ to the 2-category of $\mathbf{B}U(1)$-gerbes on $X$. Now, we have to convince ourselves that this 2-category is a group-object and so a 3-group. But this reduces to saying how to multiply two $\mathbf{B}U(1)$-gerbes (and to invert one), and this is accomplished by looking at transition functions, exactly as for principal $U(1)$-bundles: in the gerbe case, transition functions are principal $U(1)$-bundles and we use the abelian 2-group structure on these to multiply and invert them. In down to earth terms, we take the tensor product of the transition line bundles of the gerbe.

One sees that this procedure iterates: we now have an abelian 3-group $\mathbf{B}^2U(1)$ and can consider principal $\mathbf{B}^2U(1)$-bundles; this will give the abelian 4-group $\mathbf{B}^3U(1)$, and so on.

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