Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of 'degroupoidification': a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements.
Deep Dive into Groupoidification Made Easy.
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the
'Groupoidification' is an attempt to expose the combinatorial underpinnings of linear algebra -the hard bones of set theory underlying the flexibility of the continuum. One of the main lessons of modern algebra is to avoid choosing bases for vector spaces until you need them. As Hermann Weyl wrote, "The introduction of a coordinate system to geometry is an act of violence". But vector spaces often come equipped with a natural basis -and when this happens, there is no harm in taking advantage of it. The most obvious example is when our vector space has been defined to consist of formal linear combinations of the elements of some set. Then this set is our basis. But surprisingly often, the elements of this set are isomorphism classes of objects in some groupoid. This is when groupoidification can be useful. It lets us work directly with the groupoid, using tools analogous to those of linear algebra, without bringing in the real numbers (or any other ground field).
For example, let E be the groupoid of finite sets and bijections. An isomorphism class of finite sets is just a natural number, so the set of isomorphism classes of objects in E can be identified with N. Indeed, this is why natural numbers were invented in the first place: to count finite sets. The real vector space with N as basis is usually identified with the polynomial algebra R[z], since that has basis z 0 , z 1 , z 2 , . . . . Alternatively, we can work with infinite formal linear combinations of natural numbers, which form the algebra of formal power series, R [[z]]. So, formal power series should be important when we apply the tools of linear algebra to study the groupoid of finite sets.
Indeed, formal power series have long been used as ‘generating functions’ in combinatorics [21]. Given a combinatorial structure we can put on finite sets, its generating function is the formal power series whose nth coefficient says how many ways we can put this structure on an n-element set. André Joyal formalized the idea of ‘a structure we can put on finite sets’ in terms of espèces de structures, or ‘structure types’ [6,14,15]. Later his work was generalized to ‘stuff types’ [4], which are a key example of groupoidification.
Heuristically, a stuff type is a way of equipping finite sets with a specific type of extra stuff -for example a 2-coloring, or a linear ordering, or an additional finite set. Stuff types have generating functions, which are formal power series. Combinatorially interesting operations on stuff types correspond to interesting operations on their generating functions: addition, multiplication, differentiation, and so on. Joyal’s great idea amounts to this: work directly with stuff types as much as possible, and put off taking their generating functions. As we shall see, this is an example of groupoidification.
To see how this works, we should be more precise. A stuff type is a groupoid over the groupoid of finite sets: that is, a groupoid Ψ equipped with a functor v : Ψ → E. The reason for the funny name is that we can think of Ψ as a groupoid of finite sets ’equipped with extra stuff’. The functor v is then the ‘forgetful functor’ that forgets this extra stuff and gives the underlying set.
The generating function of a stuff type v : Ψ → E is the formal power series
Here v -1 (n) is the ’essential inverse image’ of any n-element set, say n ∈ E. We define this term later, but the idea is straightforward: v -1 (n) is the groupoid of n-element sets equipped with the given type of stuff. The nth coefficient of the generating function measures the size of this groupoid. But how? Here we need the concept of groupoid cardinality. It seems this concept first appeared in algebraic geometry [5,16]. We rediscovered it by pondering the meaning of division [4]. Addition of natural numbers comes from disjoint union of finite sets, since
Multiplication comes from cartesian product:
If a group G acts on a set S, we can ‘divide’ the set by the group and form the quotient S/G. If S and G are finite and G acts freely on S, S/G really deserves the name ‘quotient’, since then
Indeed, this fact captures some of our naive intuitions about division. For example, why is 6/2 = 3? We can take a 6-element set S with a free action of the group G = Z/2 and construct the set of orbits S/G: Since we are ‘folding the 6-element set in half’, we get |S/G| = 3.
The trouble starts when the action of G on S fails to be free. Let’s try the same trick starting with a 5-element set:
We don’t obtain a set with 2 1 2 elements! The reason is that the point in the middle gets mapped to itself. To get the desired cardinality 2 1 2 , we would need a way to count this point as ‘folded in half’.
To do this, we should first replace the ordinary quotient S/G by the ‘action groupoid’ or ‘weak quotient’ S//G. This is the groupoid where objects are elements of S, and a morphism from s ∈ S to s ′ ∈ S is an element g ∈ G with gs = s ′ . Composition of morphisms works in the ob
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