A strict action of a strict 2-group (seen as 2-category with only one object $\star$) $\mathcal{G}$ on a 2-space (principally a category) $\mathcal{M}$ is a strict 2-functor $\Phi:\mathcal{G}\to \mathsf{DiffCat}$ with:
- $\Phi(\star) = \mathcal{M}$
- $\forall g\in\mathcal{G}^{1}: \Phi_{g}:=\Phi(g)$ is a smooth map (principally an endofunctor) $\Phi_{g}:\mathcal{M}\to\mathcal{M}$
- $\forall h\in\mathcal{G}^{2}$ such that $h:g\Rightarrow g'$, $\Phi_{h}:=\Phi(h)$ is a smooth 2-map (principally a natural transformation) $\Phi_{h}:\Phi_{g}\Rightarrow\Phi_{g'}$
(see for example Transformation Double Categories Associated to 2-Group Actions Def. 3.2)
The automorphism 2-group of $\mathcal{M}$ is, as pointed out in Higher Gauge Theory (page 15), a coherent 2-group $\mathcal{AUT}(\mathcal{M})$ whose morphisms are (smooth) autoequivalences of $\mathcal{M}$, and whose 2-morphisms are (smooth) invertible 2-maps. An autoequivalence $t$ (with weak inverse $\bar{t}$) is less strict than an (strictly) invertible (smooth) functor, in the sens that the strict relation $t\bar{t}=\bar{t}t=id_{\mathcal{M}}$ becomes a mere (smooth, invertible) pair of natural transformations $\tau:t\bar{t}\Rightarrow id_{\mathcal{M}}$ and $\bar{\tau}:\bar{t} t\Rightarrow id_{\mathcal{M}}$.
As I see it, the image of a strict action $\Phi$ as defined above contains not only strictly invertible endofunctors, but also potentially autoequivalences:
- The existence of strict inverses is ensured by the 2-group action axioms:
$$\Phi_{gg'}=\Phi_{g}\circ\Phi_{g'}$$
which gives:
$$\Phi_{g}\circ\Phi_{g^{-1}} = \Phi_{e}=id_{\mathcal{M}}$$
- The existence of weak inverses is implied by the existence of some 2-morphisms of the form:
$$h:gg'\Rightarrow 1$$
which gives invertible natural transformations:
$$\Phi_{h}:\Phi_{g}\circ\Phi_{g'}\Rightarrow id_{\mathcal{M}}$$
The sctrict inverses will always exist, but the existence of the weak inverses is liable to the existence of 2-morphisms of the form $h:gg'\Rightarrow 1$.
Let $p:E\to B$ a locally trivializable bundle with a cover $\{U_i\}$ of $B$ and typical fiber $F$ (restricting to a trivial base 2-space for the sake of simplicity).
The local triviality of this bundle is encoded in the existence of the following diagram which defines local trivializations $t_i$, and which lives in $\mathsf{Set}$:
and which give the relation:
$$\pi_1\circ t_i = p$$
since the canonical 2-morphisms of $\mathsf{Set}$ (if considered as a 2-category) are all identities.
The categorification of the concept of locally trivializable bundle (the so called 2-bundle) lies (as pointed out in From Loop Space Mechanics to Nonabelian Strings page 46) in the replacement of the above diagram by a diagram living not in $\mathsf{Set}$ but in $\mathsf{Cat}$ (or $\mathsf{DiffCat}$ for smoothness), and since $\mathsf{Cat}$, if considered as a 2-category, possesses non-trivial natural transformations, the precedent equality becomes a mere natural transformation, in the sens that $t_i$ are in fact equivalences with weak inverses $\bar{t_i}:U_i \times F \to E|_{U_i}$, that is, there exists natural transformations $\tau$ and $\bar{\tau}$ such that:
$$\tau:t_i\bar{t_i}\Rightarrow 1$$ $$\bar{\tau}:\bar{t_i}t_i\Rightarrow 1$$
In particular, this means that $\bar{t}$ is also an equivalence, which permits us to construct autoequivalences for double overlaps $U_{ij}$ (Higher Gauge Theory):
$$t_j\bar{t_i}:U_{ij}\times F\to U_{ij}\times F$$
Since this autoequivalence acts trivially on the $U_{ij}$ factor, it will define a function $g_{ij}:U_{ij}\to\mathcal{Aut}(F)$
As defined in Higher Gauge Theory (page 19), a locally trivial 2-bundle $p:E\to B$ is said to have $\mathcal{G}$ as its structure 2-group when, in particular, the transition functions $g_{ij}$ factor through an action $\Phi:\mathcal{G}\to\mathcal{Aut}(F)$
My questions
- Is all what I said coherent?
- Does a s̲t̲r̲i̲c̲t̲ action of a s̲t̲r̲i̲c̲t̲ 2-group on a category (or a 2-space) give, inter alia, a̲u̲t̲o̲e̲q̲u̲i̲v̲a̲l̲e̲n̲c̲e̲s̲? Really?
- If yes, not every 2-group can be the structure 2-group of a given 2-bundle (like for a 1-bundle in fact), but with a higher order condition, namely, the existence of a 2-morphism $h:gg'\Rightarrow 1$ for every pair of morphisms $(g,g')$ that happens to give an autoequivalence (via the 2-group action) on some path $U_i$
old question
I'm stuggling with an issue I found in Baez and Schreiber article "Higher Gauge Theory". They say (pp. 9 line 1)
For the benefit of experts, we should admit that we are only defining ‘strict’ Lie 2-groups
and (definition 16):
A (strict) action of a smooth 2-group $\mathcal{G}$ on a smooth 2-space $F$ is a smooth homomorphism,
$\alpha: \mathcal{G}\rightarrow \mathcal{AUT}(F)$
that is, a smooth map that preserves products and inverses.
However, they also said (pp. 15):
The 2-space $\mathcal{AUT}(F)$ is a kind of a 2-group, with composition of autoequivalences giving the product. However, it is not the sort of 2-group we have been considering here, because it does not have 'strict inverses': the group laws involving inverses do not hold as equations, but only up to specified isomorphisms that satisfy coherence laws of their own. So, $\mathcal{AUT}(F)$ is a 'coherent' smooth Lie group in the sense of Baez and Lauda [12].
My question: the functoriality of the action $\alpha$ should give strict inverses for endofunctors of $F$, not weak ones, because morphisms of $\mathcal{G}$ are in fact isomorphisms (they have strict inverses). Why did these authors say that the action functor will give autoequivalences of $F$? Is there something I missed?
