Yoneda Simplification

In this section, we point out a simplification of the additive structure of A, based on Yoneda’s Lemma. That lemma (see [7, Sect. 3.2]) says roughly that an object in any category is determined, up to isomorphism, by the morphisms into it. More precisely, any category C is equivalent to a full subcategory of the category C of contravariant functors from C to the category of sets.[1] Under this equivalence, any object X of C corresponds to the functor Hom(-, X), i.e., the functor sending each object U of C to the set of morphisms U ^ X and sending each morphism f : U ^ V to the operation Hom(V, X) ^ Hom(U, X) of composition with f.

In the case of our category A, we can greatly simplify A while still maintaining the Yoneda equivalence. In the first place, since every object U of A is a finite sum, and thus in particular a coproduct, of simple objects, U = ® ?eF Sj, morphisms U ^ X amount to F-indexed families of morphisms Sj ^ X. More precisely, any f : U ^ X is determined by the composite morphisms f о Uj : Sj ^ X, and, conversely, any family of morphisms gj : Sj ^ X arises in this way from a unique morphism U ^ X. Thus, A is equivalent to a full subcategory of the category A of set-valued functors on the category A of simple objects in A.

Up to equivalence, we need not use all the simple objects; it suffices to have at least one representative from each isomorphism class of simple objects. So we can replace the A of the preceding paragraph by a skeleton of it, i.e., a full subcategory A0 consisting of just one representative per isomorphism class.

The structure of this new, skeletal A0 admits, thanks to the finiteness axiom and Proposition 4 the following description. There are finitely many objects. The morphisms from any object to itself form a copy of C. If U and V are distinct objects, then the only morphism from U to V is zero.

As a result, the Yoneda embedding, simplified as above, sends each object X of A to a finite family of vector spaces, indexed by the simple objects U in A0, namely the vector spaces Hom(U, X). Furthermore, the morphisms X ^ Y in A are given by arbitrary families of linear maps gU : Hom(U, X) ^ Hom(U, Y) between corresponding vector spaces. The reason for “arbitrary” is that, because of the paucity of morphisms in A0, all such families automatically satisfy the commutativity conditions required in order to be natural transformations and thus to be morphisms in the functor category A0.

Summarizing, we have that, up to equivalence of categories, A can be described as the category whose objects (resp. morphisms) are families of finite-dimensional vector spaces (resp. linear maps), indexed by the objects of A0. Furthermore, it is easy to check that sums in A are given, via this equivalence, by direct sums of vector spaces.

In other words, the additive structure of A is trivial. The interesting structure is the monoidal structure, and this can be quite complicated. In particular, the associativity isomorphisms a and the braiding isomorphisms a, though given (like any morphisms) by linear maps, need not have a particularly simple structure.

The analysis of the multiplicative structure of A can be facilitated by taking advantage of the semisimplicity of A and the fact that ® distributes over ®. If we know how ® acts on simple objects, distributivity determines how it acts on sums of simple objects, and, by semisimplicity, those are all the objects. Moreover, because the associativity and braiding isomorphisms are natural, and thus in particular commute with the injection and projection morphisms of sums, the behavior of these isomorphisms on arbitrary objects is determined by their behavior on simple objects. Better yet, the pentagon and hexagon conditions will be satisfied in general as soon as they are satisfied for simple objects.

Thus, the additive and multiplicative structure of A can be completely described by giving

  • 1. a complete list of non-isomorphic simple objects (including the unit or vacuum 1),
  • 2. for each pair of objects in this list, their ^-product, expressed as a sum of objects from the list,
  • 3. the associativity isomorphisms aX,Y, Z for all X, Y, Z in the list, and
  • 4. the braiding isomorphisms aX,Y for all X, Y in the list,

subject to the pentagon and hexagon conditions.

We shall not be concerned here with duality and ribbon structure, but it could also be reduced to a consideration of the simple cases.

Often, items (1) and (2) here determine or at least greatly constrain items (3) and (4) via the pentagon and hexagon conditions. One such situation is the subject of the next section. Other examples, both of strong constraints on (3) and (4) and of weak constraints can be found in [2].

  • [1] There are set-theoretic issues if C is a proper class rather than a set, but these issues need notconcern us here. The finiteness conditions imposed on our anyon category A ensure that it isequivalent to a small, i.e., set-sized, category.
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