Time once more to take stock, stop throwing out half-cooked ideas, and be a bit more careful about what I say.

I want to discuss various special cases of 2-spaces and their 2-groups.

1) A skeletal 2-space.

Here we take an ordinary space of points, $B,$ in some category $C$ of spaces, with an automorphism group of spatial transformations, $Aut(B),$ and an endomorphism monoid of spatial transformations, $End(B),$ and then we turn each point of this space into a group. The members of the latter group are to count as the morphisms of the 2-space $S$. No two points are isomorphic to one another under the 2-space morphisms, although of course they may be isomorphic as groups.

Typically we want the totality of morphisms, $M,$ to be a space of the same type as $B.$ Moreover, we’d like the group of morphisms over any individual point $p$, which we can call $A_p,$ to be a subspace of the space $M$ of all morphisms, that is, to be isomorphic to some group object in $C.$ Thus, among other requirements, we want left and right multiplication in one of the groups $A_p$ to be members of $Aut(A_p),$ that is an automorphism of that space in the category $C.$

Functors from the 2-space to itself are endomorphisms of $B$ (as an object in $C)$, which also act as homomorphisms from the group over any point in $B$ to the group over its image, and which are also morphisms (in the category $C$) of the groups over the points considered as spaces (i.e. as objects in $C.)$ Since no two objects are isomorphic as points, a natural transformation between two functors exists only if the functors have the same action as each other when considered as endomorphisms of $B$, and differ by an inner automorphism in their action on the group over each point, $A_p.$

We’d like to consider a functor from $S$ to itself to be an automorphism of $S$ iff it’s invertible up to equivalence, which requires it to act as an automorphism of $B$, i.e. as a member of $Aut(B)$ in $C$, and also as an isomorphism from each $A_p$ to the group over the image of $p,$ (that is, $A_{f(p)},$ where $f$ is the function given by the action of the functor on objects of $S,$ i.e. on points in $B.)$ Since $S$ is skeletal, the automorphisms of $S$, together with their equivalences, form a *strict* 2-group.

If all the groups $A_p$ are isomorphic (i.e. for all $p$) then the automorphism 2-group is relatively straightforward. It is $Hom(B,A)\rightarrow Aut(B)\rtimes Hom(B,Aut(A)).$ Here, the arrow (the “$\delta$” arrow) takes a map $f$ in $Hom(B, A)$ to the action which is the identity on $B$, and is the composition $inn \circ f$ on $Hom(B, Aut(A))$, where $inn$ takes a member $a$ of $A$ to conjugation by $a.$ The action of $Aut(B)\rtimes Hom(B,Aut(A)$ on $Hom(B,A)$ involves the natural action of $Aut(B)$ on $B$ as the source of maps in $Hom(B, A),$ and the natural action of $Aut(A)$ on the target $A$, after composition with $Aut(B)$ acting on the location of the point.

Now, as far as automorphisms are concerned, we have a kind of fibre bundle over $B,$ in which the fibre is an $Aut(A)$ torsor (not an $A$-torsor as I erroneously said elsewhere). Any member of $Aut(B)$ that takes a point $p$ to a point $q$ constitutes a path between $p$ and $q$. The $Aut(A)$ part of the automorphism, over $p,$ (or $q$; I think either convention works) gives the value of a connection over this path. As the match of $Aut(A)$-torsor to the group $Aut(A)$ is a matter of convention, we can twiddle the latter to our heart’s content, producing passive gauge transformations.

Confusingly, gauge changes match up functors that agree on the base space, by conjugating their action at each point by a member of $Aut(A)$. So the (trivialisations of the) functors are related by inner automorphisms in $Aut(A).$ This is *not* the same as equivalence of functors, which relates functors by composition with an inner automorphism in $A.$

This is so confusing, I’m not 100% sure I believe it, but there doesn’t seem to be a mistake that I can see.

We should also consider the case where the groups $A_p$ over points $p$ are not all the same as each other. Since automorphisms of the 2-space $S$ have to act as isomorphisms of the groups $A_p,$, this causes the base space $B$ to fall apart into isomorphism classes, that is, sets of points that are isomorphic as groups. This will cut down the choice of automorphisms of the base space $B$ to some subgroup $X$ of $Aut(B),$ namely the subgroup which preserves each of the isomorphism classes of points. In turn, this subgroup may not act transitively on each isomorphism class, causing these classes to fall apart further into smaller classes. However, each of these smaller classes will then be a skeletal 2-space in its own right. Its base space automorphisms will form the subgroup $X$ of $Aut(B),$ or possibly some subgroup of this if $X$ multiply covers the automorphism group of the base space of the subspace (i.e. the homomorphism of $X$ to this group has non-trivial kernel).

In other respects, therefore, in these circumstances, the space $S$ can simply be considered a rather heterogeneous collection of skeletal 2-spaces, on which the same strict 2-group happens to act. As far as automorphisms are concerned, I don’t think there is anything new here that isn’t already present in the case that all points are isomorphic as groups. Endomorphisms in these circumstances are a great deal more complicated, and possibly not very enlightening, so I shall avoid talking about them for the moment (or possibly ever).

We now need to consider sub-2-groups of the 2-group $Aut(S).$ Since we want to think of these as “injective” (in some sense) morphisms (2-functors) into $Aut(S),$ I want first of all to consider automorphisms of $Aut(S).$

Once again, I want to consider an automorphism of $Aut(S)$ to be a functor from $Aut(S)$ to itself which is invertible up to equivalence.

Let us consider momentarily the case where the base space $B$ of the 2-space $S$ has just one point, which reduces the automorphism 2-group of $S$ to the familiar case $A\rightarrow Aut(A).$

In this case, an automorphism of the 2-group consists of an endomorphism of $Aut(A)$ with various properties, such as that its kernel is a subgroup of $Inn(A).$ (That ensures that two objects only get sent to the same object if they are equivalent.) To track down these properties, let us start by thinking about strict automorphisms, that is compatible automorphisms of $A$ and $Aut(A).$

The 2-group $A\rightarrow Aut(A)$ is not merely a category but of course a groupoid. In this groupoid, the objects of the component containing the identity of $Aut(A)$ are precisely the elements of $Inn(A),$ the subgroup of inner automorphisms of $A.$ Each arrow in the groupoid gets labelled with a member of $A,$ such that the target of the arrow is obtained from its source by multiplying the latter on the left (in $Aut(A))$ by the inner automorphism corresponding to the label (from $A)$ on the arrow. Each member of $Aut(A)$ acquires a group of automorphisms by virtue of being an object in a groupoid. This group of automorphisms is isomorphic to the kernel of the $\delta$ map $inn,$ in other words to the centre of $A,$ $Z(A).$

Strict automorphisms of $A\rightarrow Aut(A)$ are functorial isomorphisms from this groupoid to itself which are also group automorphims of $Aut(A)$ (the set of objects) and of $A$ (the set of morphism templates). In particular, since the identity component of the groupoid must be preserved, the group of inner automorphisms $Inn(A)$ must be sent to itself.

Inner automorphisms of $Aut(A)$ (*not* inner automorphisms of $A!)$ have the right properties to give automorphisms of $A\rightarrow Aut(A),$ since $Inn(A)$ is a normal subgroup of $Aut(A),$ (hence preserved by inner automorphisms…) and since conjugation of the object group $Aut(A)$ by a member $a$ of $Aut(A)$ lifts nicely to the automorphism of the arrow group $A$ consisting precisely of the action $a.$

Since this construction uses up all possible automorphisms of the morphism group $A,$ I guess it exhausts the possible actions of automorphisms of $Aut(A)$ on $Inn(A).$ There is also the potential for automorphisms of $Aut(A)$ not to preserve $Inn(A),$ and hence not to be functors, so obviously these need to be excluded. However, I don’t see any obstacle to including outer automorphisms of $Aut(A)$ which act like inner automorphisms when restricted to $Inn(A),$ but do something non-trivial to $Out(A).$ I haven’t thought too much about whether this makes sense, though.

Two of these strict automorphisms will be equivalent if they agree on how they permute the components of the groupoid among themselves, which I think amounts to differing by an inner automorphism of $Aut(A)$ (together with its lift to $A).$

These strict automorphisms of $A\rightarrow Aut(A)$ will induce corresponding automorphisms of $A$, the fibre of our skeletal 2-space $S,$ in order to preserve the action. Of course, these don’t select any exciting figures or anything like that.

Now, so far we’ve only considered “strict” automorphisms of $A\rightarrow Aut(A).$ However, although $A\rightarrow Aut(A)$ is a *strict* 2-group, it isn’t (in general) a *skeletal* 2-group, i.e. the components of its groupoid structure have more than one object. We therefore have the possibility of “weak” automorphisms, which are not injective on objects. The crucial requirement, I think, is that endomorphisms of $Aut(A)$ have to send $Inn(A)$ to itself, endomorphically if not automorphically. This ensures that the map is a functor on the groupoid. If objects are merged, then of course morphism templates will be merged as well. In the extreme case, our “essentially surjective 2-subgroup” will be a skeletal strict 2-group, and strictly isomorphic to $Z(A)\rightarrow Out(A),$ with $\delta$-map zero. However, depending on which endomorphism we choose, we may end up with different possible realisations of this skeletal 2-group as actual automorphisms of $A.$

In intermediate cases, where $Inn(A)$ is reduced but not shrunk down to $0,$ the details of what endomorphisms are and are not allowed can get quite tricky, I think, because $Out(A)$ still needs to have a valid action on the image of $Inn(A),$ which won’t be the case for every endomorphism of $Inn(A).$ I haven’t examined these restrictions in detail, though it might be interesting to work out what is going on here – particularly since this is where key parts of the “2-ness” of 2-spaces would appear to reside.

The cut-down automorphism 2-group can be seen as defining a “figure” in the group $A$. I’m not clear how one should best view these figures. One can of course construct the orbit of each member of $A$ under the action of a given 2-group of automorphisms. A more “groupy” way of thinking about this would be to consider the subgroups of $A$ generated by these orbits. Or maybe one should think about sublattices of the subgroup lattice of $A.$

One possible rather 2-spacey way to think about it is to consider that, given any subgroup of $Aut(A)$, the subgroups of $A$ form a groupoid, with the subgroups of $A$ as objects, and isomorphisms existing between two subgroups iff they are isomorphically mapped to one another by some action of the subgroup of $Aut(A)$ that we are considering. Different subgroups of $Aut(A)$ will give different subgroup subgroupoids of the complete groupoid of $Aut(A)$ (which is itself a subgroupoid of the groupoid of subgroups within **Grp**). We can lump subgroups of $Aut(A)$ together if they give rise to the same groupoid, and think about the partial ordering of groupoids by inclusion.

Anyway, that’s all I have to say for the moment about the case of a base space with just one point.

If we now move to a base space $B$ with more than one point, we have to consider the possibility that we might be allowed to assign different essentially surjective 2-subgroups of $A\rightarrow Aut(A)$ to different points of $B.$ I don’t think there is any actual obstacle to doing this (the group $A$ over each point is still the same) but it does lead to the odd possibility that one may be able to perform an automorphism on the group at a given point by taking it around a path over $B,$ even though one cannot perform the same automorphism keeping the group in place.

Well, I guess this isn’t too odd: one can invert a frame by taking it around a path in a non-orientable manifold, even if one can’t invert it locally, and similarly for other phenomena in twisted bundles.

It’s at this point that I ought to begin talking about actual non-essentially-surjective 2-subgroups of, at least, $A\rightarrow Aut(A).$ However, at 2000 words, I think this post is more than long enough already, and I am growing weary. The only remark I have ready to hand to make at this point is that subgroups which are strictly surjective on the object group $Aut(A),$ but cut down the morphism template group $A,$ seem rather straightforward. The real interest arises from the interaction between the proper subgroups of each of these groups. For instance, when removing objects, one has to be careful that they are not accidentally “regenerated” by the action of the morphism templates, springing back like some infuriating weed.

One other remark I have to make, on a matter that I haven’t really thought about too hard, is that if we define 2-subgroups in terms of 2-equivalence classes of 2-groups that can be 2-functorially mapped into a given 2-group, then one finds oneself allowing arbitrarily large object groups to map into the target object group! This is because injectivity is allowed to fail whenever two objects in the source are isomorphic (under morphisms of the source). This is rather a peculiar situation. Whatever interest it may hold, I suspect lies again in the interaction between subgroups and homomorphisms of morphisms, and subgroups and homomorphisms of objects. Perhaps this is a rather trite observation, but it suggests an area to focus my attention.

This discussion could do with an extended example. I have been working one out as an adjunct to writing this post, so perhaps in my next post I shall work this up into something readable, maybe even with pretty pictures if I can work out how to make this happen.

Then maybe a bit about various kinds of non-surjective 2-subgroups, and then I’ll look at another kind of 2-space.

## Re: Klein 2-Geometry VI

I’m afraid I’m still too distracted to push this Klein 2-geometry project forwards very quickly. But let me give it a nudge.

David wrote:

That sounds right to me. We’re going by the seat of our pants here, but I think it all works out quite simply.

In fact my main worry now is that it’s too simple to be very interesting! We’ll need to ask some tougher questions about these Grassmannian to see if they have hidden depths. For example, let’s study the “incidence relations” between the $(n,0)$ sub-2-spaces and $(m,0)$ sub-2-spaces of $k^{p,q}$. Then we’ll finally be doing categorified incidence geometry.

Of course we’re using enough jargon and symbols now that nobody except us can possibly understand what we’re talking about. So, my claim that it’s “too simple” may strike them as unconvincing.

But maybe that’s how math advances: when you’re completely stuck and a problem seems incomprehensible, lots of people can understand what you’re whining about. But when you finally crack the problem and it seems completely lucid, nobody can understand you anymore!

To reach understanding it seems one must suffer confusion, even when there’s somebody standing there telling you anything you want to know. The neurons must be reconfigured.