Categorifying Global vs. Local Symmetry

So it’s been a while since I last posted – the end of 2013 ended up being busy with a couple of visits to Jamie Vicary in Oxford, and Roger Picken in Lisbon. In the aftermath of the two trips, I did manage to get a major revision of this paper submitted to a journal, and put this one out in public. A couple of others will be coming down the pipeline this year as well.

I’m hoping to get back to a post about motives which I planned earlier, but for the moment, I’d like to write a little about the second paper, with Roger Picken.

Global and Local Symmetry

The upshot is that it’s about categorifying the concept of symmetry. More specifically, it’s about finding the analog in the world of categories for the interplay between global and local symmetry which occurs in the world of set-based structures (sets, topological spaces, vector spaces, etc.) This distinction is discussed in a nice way by Alan Weinstein in this article from the Notices of the AMS from

The global symmetry of an object $X$ in some category $\mathbf{C}$ can be described in terms of its group of automorphisms: all the ways the object can be transformed which leave it “the same”. This fits our understanding of “symmetry” when the morphisms can really be interpreted as transformations of some sort. So let’s suppose the object is a set with some structure, and the morphisms are set-maps that preserve the structure: for example, the objects could be sets of vertices and edges of a graph, so that morphisms are maps of the underlying data that preserve incidence relations. So a symmetry of an object is a way of transforming it into itself – and an invertible one at that – and these automorphisms naturally form a group $Aut(X)$ . More generally, we can talk about an action of a group $G$ on an object $X$ , which is a map $\phi : G \rightarrow Aut(X)$ .

“Local symmetry” is different, and it makes most sense in a context where the object $X$ is a set – or at least, where it makes sense to talk about elements of $X$ , so that $X$ has an underlying set of some sort.

Actually, being a set-with-structure, in a lingo I associate with Jim Dolan, means that the forgetful functor $U : \mathbf{C} \rightarrow \mathbf{Sets}$ is faithful: you can tell morphisms in $\mathbf{C}$ (in particular, automorphisms of $X$ ) apart by looking at what they do to the underlying set. The intuition is that the morphisms of $\mathbf{C}$ are exactly set maps which preserve the structure which $U$ forgets about – or, conversely, that the structure on objects of $\mathbf{C}$ is exactly that which is forgotten by $U$ . Certainly, knowing only this information determines $\mathbf{C}$ up to equivalence. In any case, suppose we have an object like this: then knowing about the symmetries of $X$ amounts to knowing about a certain group action, namely the action of $Aut(X)$ , on the underlying set $U(X)$ .

From this point of view, symmetry is about group actions on sets. The way we represent local symmetry (following Weinstein’s discussion, above) is to encode it as a groupoid – a category whose morphisms are all invertible. There is a level-slip happening here, since $X$ is now no longer seen as an object inside a category: it is the collection of all the objects of a groupoid. What makes this a representation of “local” symmetry is that each morphism now represents, not just a transformation of the whole object $X$ , but a relationship under some specific symmetry between one element of $X$ and another. If there is an isomorphism between $x \in X$ and $y \in X$ , then $x$ and $y$ are “symmetric” points under some transformation. As Weinstein’s article illustrates nicely, though, there is no assumption that the given transformation actually extends to the entire object $X$ : it may be that only part of $X$ has, for example, a reflection symmetry, but the symmetry doesn’t extend globally.

Transformation Groupoid

The “interplay” I alluded to above, between the global and local pictures of symmetry, is to build a “transformation groupoid” (or “action groupoid“) associated to a group $G$ acting on a set $X$ . The result is called $X // G$ for short. Its morphisms consist of pairs such that $(g,x) : x \rightarrow (g \rhd x)$ is a morphism taking $x$ to its image under the action of $g \in G$ . The “local” symmetry view of $X // G$ treats each of these symmetry relations between points as a distinct bit of data, but coming from a global symmetry – that is, a group action – means that the set of morphisms comes from the product $G \times X$ .

Indeed, the “target” map in $X // G$ from morphisms to objects is exactly a map $G \times X \rightarrow X$ . It is not hard to show that this map is an action in another standard sense. Namely, if we have a real action $\phi : G \rightarrow Hom(X,X)$ , then this map is just $\hat{\phi} : G \times X \rightarrow X$ , which moves one of the arguments to the left side. If $\phi$ was a functor, then $\hat{\phi}$ satisfies the “action” condition, namely that the following square commutes:

(Here, $m$ is the multiplication in $G$ , and this is the familiar associativity-type axiom for a group action: acting by a product of two elements in $G$ is the same as acting by each one successively.

So the starting point for the paper with Roger Picken was to categorify this. It’s useful, before doing that, to stop and think for a moment about what makes this possible.

First, as stated, this assumed that $X$ either is a set, or has an underlying set by way of some faithful forgetful functor: that is, every morphism in $Aut(X)$ corresponds to a unique set map from the elements of $X$ to itself. We needed this to describe the groupoid $X // G$ , whose objects are exactly the elements of $X$ . The diagram above suggests a different way to think about this. The action diagram lives in the category $\mathbf{Set}$ : we are thinking of $G$ as a set together with some structure maps. $X$ and the morphism $\hat{\phi}$ must be in the same category, $\mathbf{Set}$ , for this characterization to make sense.

So in fact, what matters is that the category $X$ lived in was closed: that is, it is enriched in itself, so that for any objects $X,Y$ , there is an object $Hom(X,Y)$ , the internal hom. In this case, it’s $G = Hom(X,X)$ which appears in the diagram. Such an internal hom is supposed to be a dual to $\mathbf{Set}$ ‘s monoidal product (which happens to be the Cartesian product $\times$ ): this is exactly what lets us talk about $\hat{\phi}$ .

So really, this construction of a transformation groupoid will work for any closed monoidal category $\mathbf{C}$ , producing a groupoid in $\mathbf{C}$ . It may be easier to understand in cases like $\mathbf{C}=\mathbf{Top}$ , the category of topological spaces, where there is indeed a faithful underlying set functor. But although talking explicitly about elements of $X$ was useful for intuitively seeing how $X//G$ relates global and local symmetries, it played no particular role in the construction.

Categorify Everything

In the circles I run in, a popular hobby is to “categorify everything“: there are different versions, but what we mean here is to turn ideas expressed in the world of sets into ideas in the world of categories. (Technical aside: all the categories here are assumed to be small). In principle, this is harder than just reproducing all of the above in any old closed monoidal category: the “world” of categories is $\mathbf{Cat}$ , which is a closed monoidal 2-category, which is a more complicated notion. This means that doing all the above “strictly” is a special case: all the equalities (like the commutativity of the action square) might in principle be replaced by (natural) isomorphisms, and a good categorification involves picking these to have good properties.

(In our paper, we left this to an appendix, because the strict special case is already interesting, and in any case there are “strictification” results, such as the fact that weak 2-groups are all equivalent to strict 2-groups, which mean that the weak case isn’t as much more general as it looks. For higher $n$ -categories, this will fail – which is why we include the appendix to suggest how the pattern might continue).

Why is this interesting to us? Bumping up the “categorical level” appeals for different reasons, but the ones matter most to me have to do with taking low-dimensional (or -codimensional) structures, and finding analogous ones at higher (co)dimension. In our case, the starting point had to do with looking at the symmetries of “higher gauge theories” – which can be used to describe the transport of higher-dimensional surfaces in a background geometry, the way gauge theories can describe the transport of point particles. But I won’t ask you to understand that example right now, as long as you can accept that “what are the global/local symmetries of a category like?” is a possibly interesting question.

So let’s categorify the discussion about symmetry above… To begin with, we can just take our (closed monoidal) category to be $\mathbf{Cat}$ , and follow the same construction above. So our first ingredient is a 2-group $\mathcal{G}$ . As with groups, we can think of a 2-group either as a 2-category with just one object $\star$ , or as a 1-category with some structure – a group object in $\mathbf{Cat}$ , which we’ll call $C(\mathcal{G})$ if it comes from a given 2-group. (In our paper, we keep these distinct by using the term “categorical group” for the second. The group axioms amount to saying that we have a monoidal category $(\mathcal{G}, \otimes, I)$ . Its objects are the morphisms of the 2-group, and the composition becomes the monoidal product $\otimes$ .)

(In fact, we often use a third equivalent definition, that of crossed modules of groups, but to avoid getting into that machinery here, I’ll be changing our notation a little.)

2-Group Actions

So, again, there are two ways to talk about an action of a 2-group on some category $\mathbf{C}$ . One is to define an action as a 2-functor $\Phi : \mathcal{G} \rightarrow \mathbf{Cat}$ . The object being acted on, $\mathbf{C} \in \mathbf{Cat}$ , is the unique object $\Phi(\star)$ – so that the 2-functor amounts to a monoidal functor from the categorical group $C(\mathcal{G})$ into $Aut(\mathbf{C})$ . Notice that here we’re taking advantage of the fact that $\mathbf{Cat}$ is closed, so that the hom-“sets” are actually categories, and the automorphisms of $\mathbf{C}$ – invertible functors from $\mathbf{C}$ to itself – form the objects of a monoidal category, and in fact a categorical group. What’s new, though, is that there are also 2-morphisms – natural transformations between these functors.

To begin with, then, we show that there is a map $\hat{\Phi} : \mathcal{G} \times \mathbf{C} \rightarrow \mathbf{C}$ , which corresponds to the 2-functor $\Phi$ , and satisfies an action axiom like the square above, with $\otimes$ playing the role of group multiplication. (Again, remember that we’re only talking about the version where this square commutes strictly here – in an appendix of the paper, we talk about the weak version of all this.) This is an intuitive generalization of the situation for groups, but it is slightly more complicated.

The action $\Phi$ directly gives three maps. First, functors $\Phi(\gamma) : \mathbf{C} \rightarrow \mathbf{C}$ for each 2-group morphism $\gamma$ – each of which consists of a function between objects of $\mathbf{C}$ , together with a function between morphisms of $\mathbf{C}$ . Second, natural transformations $\Phi(\eta) : \Phi(\gamma) \rightarrow \Phi(\gamma ')$ for 2-morphisms $\eta : \gamma \rightarrow \gamma'$ in the 2-group – each of which consists of a function from objects to morphisms of $\mathbf{C}$ .

On the other hand, $\hat{\Phi} : \mathcal{G} \times \mathbf{C} \rightarrow \mathbf{C}$ is just a functor: it gives two maps, one taking pairs of objects to objects, the other doing the same for morphisms. Clearly, the map $(\gamma,x) \mapsto x'$ is just given by $x' = \Phi(\gamma)(x)$ . The map taking pairs of morphisms $(\eta,f) : (\gamma,x) \rightarrow (\gamma ', y)$ to morphisms of $\mathbf{C}$ is less intuitively obvious. Since I already claimed $\Phi$ and $\hat{\Phi}$ are equivalent, it should be no surprise that we ought to be able to reconstruct the other two parts of $\Phi$ from it as special cases. These are morphism-maps for the functors, (which give $\Phi(\gamma)(f)$ or $\Phi(\gamma ')(f)$ ), and the natural transformation maps (which give $\Phi(\eta)(x)$ or $\Phi(\eta)(y)$ ). In fact, there are only two sensible ways to combine these four bits of information, and the fact that $\Phi(\eta)$ is natural means precisely that they’re the same, so:

$\hat{\Phi}(\eta,f) = \Phi(\eta)(y) \circ \Phi(\gamma)(f) = \Phi(\gamma ')(f) \circ \Phi(\eta)(x)$

Given the above, though, it’s not so hard to see that a 2-group action really involves two group actions: of the objects of $\mathcal{G}$ on the objects of $\mathbf{C}$ , and of the morphisms of $\mathcal{G}$ on objects of $\mathbf{C}$ . They fit together nicely because objects can be identified with their identity morphisms: furthermore, $\Phi$ being a functor gives an action of $\mathcal{G}$ -objects on $\mathbf{C}$ -morphisms which fits in between them nicely.

But what of the transformation groupoid? What is the analog of the transformation groupoid, if we repeat its construction in $\mathbf{Cat}$ ?

The Transformation Double Category of a 2-Group Action

The answer is that a category (such as a groupoid) internal to $\mathbf{Cat}$ is a double category. The compact way to describe it is as a “category in $\mathbf{Cat}$ “, with a category of objects and a category of morphisms, each of which of course has objects and morphisms of its own. For the transformation double category, following the same construction as for sets, the object-category is just $\mathbf{C}$ , and the morphism-category is $\mathcal{G} \times \mathbf{C}$ , and the target functor is just the action map $\hat{\Phi}$ . (The other structure maps that make this into a category in $\mathbf{Cat}$ can similarly be worked out by following your nose).

This is fine, but the internal description tends to obscure an underlying symmetry in the idea of double categories, in which morphisms in the object-category and objects in the morphism-category can switch roles, and get a different description of “the same” double category, denoted the “transpose”.

A different approach considers these as two different types of morphism, “horizontal” and “vertical”: they are the morphisms of horizontal and vertical categories, built on the same set of objects (the objects of the object-category). The morphisms of the morphism-category are then called “squares”. This makes a convenient way to draw diagrams in the double category. Here’s a version of a diagram from our paper with the notation I’ve used here, showing what a square corresponding to a morphism $(\chi,f) \in \mathcal{G} \times \mathbf{C}$ looks like:

The square (with the boxed label) has the dashed arrows at the top and bottom for its source and target horizontal morphisms (its images under the source and target functors: the argument above about naturality means they’re well-defined). The vertical arrows connecting them are the source and target vertical morphisms (its images under the source and target maps in the morphism-category).

Horizontal and Vertical Slices of $\mathbf{C} // \mathcal{G}$

So by construction, the horizontal category of these squares is just the object-category $\mathbf{C}$ . For the same reason, the squares and vertical morphisms, make up the category $\mathcal{G} \times \mathbf{C}$ .

On the other hand, the vertical category has the same objects as $\mathbf{C}$ , but different morphisms: it’s not hard to see that the vertical category is just the transformation groupoid for the action of the group of $\mathbf{G}$ -objects on the set of $\mathbf{C}$ -objects, $Ob(\mathbf{C}) // Ob(\mathcal{G})$ . Meanwhile, the horizontal morphisms and squares make up the transformation groupoid $Mor(\mathbf{C}) // Mor(\mathcal{G})$ . These are the object-category and morphism-category of the transpose of the double-category we started with.

We can take this further: if squares aren’t hip enough for you – or if you’re someone who’s happy with 2-categories but finds double categories unfamiliar – the horizontal and vertical categories can be extended to make horizontal and vertical bicategories. They have the same objects and morphisms, but we add new 2-cells which correspond to squares where the boundaries have identity morphisms in the direction we’re not interested in. These two turn out to feel quite different in style.

First, the horizontal bicategory extends $\mathbf{C}$ by adding 2-morphisms to it, corresponding to morphisms of $\mathcal{G}$ : roughly, it makes the morphisms of $\mathbf{C}$ into the objects of a new transformation groupoid, based on the action of the group of automorphisms of the identity in $\mathcal{G}$ (which ensures the square has identity edges on the sides.) This last point is the only constraint, and it’s not a very strong one since $Aut(1_G)$ and $G$ essentially determine the entire 2-group: the constraint only relates to the structure of $\mathcal{G}$ .

The constraint for the vertical bicategory is different in flavour because it depends more on the action $\Phi$ . Here we are extending a transformation groupoid, $Ob(\mathbf{C}) // Ob(\mathcal{G})$ . But, for some actions, many morphisms in $\mathcal{G}$ might just not show up at all. For 1-morphisms $(\gamma, x)$ , the only 2-morphisms which can appear are those taking $\gamma$ to some $\gamma '$ which has the same effect on $x$ as $\gamma$ . So, for example, this will look very different if $\Phi$ is free (so only automorphisms show up), or a trivial action (so that all morphisms appear).

In the paper, we look at these in the special case of an adjoint action of a 2-group, so you can look there if you’d like a more concrete example of this difference.

Speculative Remarks

The starting point for this was a project (which I talked about a year ago) to do with higher gauge theory – see the last part of the linked post for more detail. The point is that, in gauge theory, one deals with connections on bundles, and morphisms between them called gauge transformations. If one builds a groupoid out of these in a natural way, it turns out to result from the action of a big symmetry group of all gauge transformations on the moduli space of connections.

In higher gauge theory, one deals with connections on gerbes (or higher gerbes – a bundle is essentially a “0-gerbe”). There are now also (2-)morphisms between gauge transformations (and, in higher cases, this continues further), which Roger Picken and I have been calling “gauge modifications”. If we try to repeat the situation for gauge theory, we can construct a 2-groupoid out of these, which expresses this local symmetry. The thing which is different for gerbes (and will continue to get even more different if we move to $n$ -gerbes and the corresponding $(n+1)$ -groupoids) is that this is not the same type of object as a transformation double category.

Now, in our next paper (which this one was written to make possible) we show that the 2-groupoid is actually very intimately related to the transformation double category: that is, the local picture of symmetry for a higher gauge theory is, just as in the lower-dimensional situation, intimately related to a global symmetry of an entire moduli 2-space, i.e. a category. The reason this wasn’t obvious at first is that the moduli space which includes only connections is just the space of objects of this category: the point is that there are really two special kinds of gauge transformations. One should be thought of as the morphisms in the moduli 2-space, and the other as part of the symmetries of that 2-space. The intuition that comes from ordinary gauge theory overlooks this, because the phenomenon doesn’t occur there.

Physically-motivated theories are starting to use these higher-categorical concepts more and more, and symmetry is a crucial idea in physics. What I’ve sketched here is presumably only the start of a pattern in which “symmetry” extends to higher-categorical entities. When we get to 3-groups, our simplifying assumptions that use “strictification” results won’t even be available any more, so we would expect still further new phenomena to show up – but it seems plausible that the tight relation between global and local symmetry will still exist, but in a way that is more subtle, and refines the standard understanding we have of symmetry today.

ubugnu Says:

January 11, 2014 at 6:30 am

Very interesting, I’m eager to see the article…

Higher Gauge Theory in Edinburgh – Part I | Theoretical Atlas Says:

August 15, 2014 at 8:38 pm

[…] of that point was described in a previous post: it has to do with finding a higher analog for the relationship between two views of symmetry: one […]

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