There’s an annoying philistinism in some areas of mathematics, including my own area of geometric group theory, of pooh-poohing category theory.
It’s easy to see why: if you browse the nLab or the Stacks project or talk to a PL theory person who knows too much Haskell, you mostly get what sounds like meaningless gobbledigook—excuse me, I meant abstract nonsense. This isn’t to bash the work that went into these projects, which are valuable references once you learn how to read them. The problem is that many people don’t discover why they might want to.
Ostensibly, the aim of a field like geometric group theory, low-dimensional topology or analysis, is concerned with answering concrete questions about fairly specific objects. Such and such family of solitons, or Artin groups or knots, or what have you. Often a strong geometric intuition or a willingness to get one’s hands dirty with the simplest nontrivial example is a key part of the work. Knowledge of category theory often does not help here. Knowing that an HNN extension is an example of a co-inserter is very different from being able to comfortably work with its Bass–Serre tree. Everyone in mathematics has met an approximation of the bright senior undergrad or first-year grad student who appears to have swallowed Category Theory for the Working Mathematician whole. This student might use abstract nonsense as a deflection, a way of avoiding actually getting their hands dirty learning a field.
That being said, there really is a time for theory-building, and geometric group theory for the most part falls on its face at this step. It’s great for definitions to be difficult so that theorems can be easy, for instance, but some definitions are still too difficult. To me, this signals a lack of time spent trying to fit tools into a bigger picture.
Anyway, that’s enough polemics from me for today. The purpose of this post is to introduce a concept familiar to folks working in category theory that suffers from a lack of attention from group theorists: the notion of a subgroupoid of a group. In an effort to make it relevant to a geometric group theorist, I’ll close by reformulating the definitions of height and width of subgroups as properties of a subgroupoid.
I’m going to fly through this because you probably have seen this stuff before, and if you haven’t, you should do more work understanding it than this post is gonna provide.
A groupoid is a small category where every arrow is invertible.
If that’s too fast for you, you have two sets of objects and of arrows. I’ll use function notation for arrows: saying that is an arrow with source object and target object . Arrows and can compose like functions as , and each object has an identity arrow . Composition is associative and identity arrows are (two-sided) identities for composition. Finally if is an arrow, there is an arrow such that and .
You might have heard “a group is a groupoid with one object.” This is true: if is a group, let itself denote the object and each element becomes an arrow . Composition is the group operation and inversion is inversion.
Morphisms of groupoids are functors: sends, for example in to in , so there are assignments of objects to objects and arrows to arrows. Composition and identities must be respected.
A natural transformation, natural isomorphism, or my emerging favorite terminology, a conjugacy between functors is an assignment, for each object , of an arrow in such that for each arrow in , the equation
is true in . The notation is .
I like the word conjugacy because in the case where and are groups, a functor is a homomorphism and a conjugacy is simply a choice of an element with the property that for all elements .
An equivalence of groupoids and is a pair of functors and and conjugacies and .
Assuming the Axiom of Choice, a functor is an equivalence if and only if it is full (surjective on arrows ), faithful (injective on arrows ) and essentially surjective (every object in the target groupoid is connected to an object by an arrow ).
A groupoid is connected if between every pair of objects and , there is an arrow . A groupoid is contractible if it is connected and the aforementioned arrow is unique. Every nonempty connected groupoid is equivalent to the group of arrows for some (and hence any) object . A nonempty contractible groupoid is equivalent to the trivial group.
If is an object, its connected component is the set of objects in such that there is an arrow . Suppose is the set of connected components of . We can construct a spanning groupoid from this set by choosing for each a unique arrow , with the stipulation that the arrow is . Completing this by addding in identities and compositions, we get a contractible groupoid with vertex set . With all of this data, one can show that each arrow factors uniquely as , where and are in and is an element of the group .
Every groupoid, then, is equivalent to what J.P. Mutanguha denotes a group system: an indexed disjoint union of groups. Explicitly, if is the set of connected components of , then is equivalent to the disjoint union of groups , where .
A subgroupoid of a group is a groupoid together with a functor with trivial kernel, meaning that if , then is an identity arrow.
Let’s analyze what this means. By our discussion in the previous section, we should consider the groups where is the set of connected components of . On each such group restricts to an injective homomorphism, so each group is a subgroup of . Each arrow maps to an element and the stipulation that no arrow (necessarily) of the form maps to the identity says that if and are distinct elements of , the cosets are distinct in .
A connected subgroupoid of a group therefore corresponds to a set of cosets of a given subgroup . The groupoid therefore has at most many objects I’ll say that a component of a subgroupoid is total if it has for its set of objects, and that is total if each component is. (I don’t love “total”. Maybe another name could be “wide”, which generally has a slightly different meaning; if you have a better name, please let me know.)
Suppose is a group. Typically should be finitely generated, perhaps even finitely presented, but the following definition makes some amount of sense for a group .
First, we say the height of an infinite subgroup is if there exist distinct cosets such that the intersection is infinite and is maximum possible.
Next, we say that the width of an infinite subgroup is if there are distinct cosets as above so that pairwise we have that and again is maximum possible.
Finite width implies finite height.
This is phrased both simply and cleanly in the language of subgroupoids: If is a total subgroupoid of , the inclusion witnesses “vertex” or “isotropy” groups as subgroups of . A component of has width (or respectively height) if is infinite and there is a collection of objects in (which are, remember, subgroups of conjugate to by conjugators from distinct cosets) with pairwise (or respectively total) infinite intersection and is chosen maximum possible. A subgroupoid therefore has width (or height) if is the maximum width (or height) among its components.