Category theory in TQFT Ethan Lake October 30, 2016 These notes are a summary of some of the aspects of category theory that play an important role in topological field theory, topological quantum computation, and so on. They follow a series of talks I gave during the spring of 2016 as part of a study group on category theory. At present, they're about 30% typed up | I may or may not come back to finish them up in the future. 1 Prerequisites from category theory 1.1 Monoidal categories and functors The program of \categorization" essentially consists of replacing equals signs by isomor- phisms. A category is a categorization of a set, an abelian category is a categorization of an abelian group, and so on. The most fundamental class of categories for us are monoidal categories, which are categorizations of monoids (recall that a monoid is a set with an associative binary operation). More precisely: Definition 1. A monoidal category is a tuple (C; ⊗; a; 1; ι) where C is an Abelian category, ⊗ : C × C ! C is a bifunctor which provides the binary operation on C, a :(− ⊗ −) ⊗ − ! − ⊗ (− ⊗ −) is a natural isomorphism called the associator, 1 is the unit object in C, and ι is an isomorphism ι : 1 ⊗ 1 ! 1. We require this data to satisfy two axioms: a) The maps R : X ⊗ 1 ! X and L : 1 ⊗ X ! X are tensor autoequivalences of C. b) The pentagon identity is satisfied. Of these, the pentagon axiom is far more important (and we won't really worry about either 1 or ι at all). A monoidal functor is a functor that preserves the monoidal structure of the categories it maps between. More precisely, a monoidal functor is a pair (F; α) with F : C!D a functor between monoidal categories C and D, and α a natural transformation (morphism of functors) that gives the tensor structure of F : αXY : F (X) ⊗D F (Y ) ! F (X ⊗C Y ): (1) 1.2 Deligne's tensor product Deligne's tensor product is a way to \put two topological phases on top of each other" which will be very useful for us later on, and which we might as well define as part of this prerequisites section. Throughout, we let C and D be two finite abelian category, as usual over some field k. 1 Definition 2. Deligne's tensor product is a bifunctor : C × D ! C D :(X; Y ) 7! X Y (2) which is universal for functors assigning to every k-linear abelian category E the category of right exact in both variables bilinear bifunctors C × D ! E. This might seem a little cryptic at first, but it's actually pretty simple. The universal- ity condition for means what it always does { is the \simplest" right exact bilinear bifunctor, meaning that all other right exact bifunctors factor through it. That is, if B is some other right exact bilinear bifunctor B : C × D ! E, then there exists a unique right exact bifunctor B : C D!E satisfying B ◦ = B. A silly analogy: If the set of all right exact bilinear bifunctors from C × D to E is the set of all ways to get from the countryside into a castle with a big moat, then is path the castle's drawbridge. Every path from the countryside into the castle must factor through the drawbridge, and so the drawbridge is `universal' for the category of paths leading into the castle. Example 1. Physically, we can interpret C D as a topological phase described by C \stacked on top of" one described by D. In particular, if we stack C on top of Cop, we in op some way \average" over the difference between X ⊗ Y and Y ⊗ X, and so C C = Z(C), where Z(C) is the center of C (which also corresponds to the quasiparticle excitations of C). This will be made more precise later on. 2 Tensor categories 2.1 Rigidity Before we define tensor categories, we need to define the notion of dual objects. This is done in pretty mucht the way we'd expect. Let C be a monoidal category, and X 2 C. We say ∗ ∗ that X is a left dual of X if there exist morphisms dX : X ⊗ X ! 1 (the death morphism) ∗ and bX : 1 ! X ⊗ X (the birth morphism) such that the maps X ! (X ⊗ X∗) ⊗ X ! X ⊗ (X∗ ⊗ X) ! X; (3) X∗ ! X∗ ⊗ (X ⊗ X∗) ! (X∗ ⊗ X) ⊗ X∗ ! X∗ are identity morphisms. This is equivalent to saying that if X has a left dual, we can remove vertical s-shaped wiggles of an X line in any diagram. The notion of a right dual ∗X is defined in the same way, but with the roles of the birth and death morphisms interchanged. The category C is called rigid if each of its objects have both left and right duals. A technical but important point: we may not always have X∗∗ = X. This is the case ∗∗ in Veck and Repk(G), but not in general. The isomorphism between X and X is called ∗∗ a pivotal structure on C, and we write it as pX : X ! X , which is natural in X and satisfies pX⊗Y = pX ⊗ pY . We need to incorporate the pivotal structure into our definition of quantum traces as the factors picked up by killing off closed bubbles in the graph. We are then led to define the pivotal trace of a morphism (not just of an object) by 1 ptrp(f : X ! X) = bX∗ (pxf ⊗ idX∗ )dX 2 EndC( ) (4) In particular, the quantum dimensions familiar from string-net diagrammatics are defined through the pivotal trance of the identity morphism: dX = ptrp(idX ) where the piv- otal structure is chosen in a canonical way so that the pivotal dimension agrees with the Frobenius-Perron dimension (to be discussed later). 2 2.2 Tensor categories, fusion categories, and tensor functors We start with some definitions. Definition 3. A multitensor category is a k-linear abelian rigid monoidal category C, where the tensor product bifunctor is bilinear on morphisms. If the unit object is simple (i.e., if EndC(1) ' k), then C is just a tensor category. If C is finite semisimple (multi)tensor category, we call C a (multi)fusion category. If we drop the assumption of rigidity, then C is called a multiring category. We will essentially always specialize to fusion categories. Some easy examples are k−Vec (fusion), Repk(G) (fusion if char k = 0 or gcd(char k; jGj) = 1), VecG, etc. One very general example of multitensor categories comes from the notion of a groupoid: Definition 4. A groupoid G is a category where all morhpisms are isomorphisms { that is, G consists of a set S of objects of G and a set G of morphisms of G, the image and preimage maps, the composition map, the unit morphism map, and the inversion map. In particular, any group can be made into a groupoid: this groupoid only has one object, and the endomorphisms of that object are given by the group G. Groupoids also provide the most general examples of multi-tensor categories. Example 2. Let G be a groupoid with a set of objects S (with S finite), and let C(G) be the category of finite dimensional vector spaces graded by the morphisms of G, which we will take to be a finite group. The objects in C(G) can thus be decomposed as S 3 X = ⊕g2GXg. The tensor product is defined naturally by M (X ⊗ Y )g = Xh ⊗ Yk (5) fh;kjhk=gg The unit object in C(G) also decomposes in a semisimple way as 1 = ⊕s2S1s. We thus see that C(G) is a multi-tensor (rather than just tensor) category. In particular, if we take S = Zn, then C(G) is a category based on \matrices of vector spaces", and is denoted by Matn(Vec). Note that this is not the same as Matn(VecG)! To get Matn(VecG), we need to consider a slightly more general scenario. Let fG^ig be the set of isomorphism classes of a groupoid G, and let Gi = Aut(gi) for some representative g of G^ . Then C(G) is (monoidally equivalent to) L Mat (Vec ). i i i jG^ij Gi Just as the definition of monoidal categories led us to define monoidal functors, tensor categories prompt us to define functors between them, which are aptly named tensor func- tors. A tensor functor isn't actually a functor (bad language!) { rather, it's a functor + natural isomorphism pair. The formal definition is as follows: Definition 5. Let F : C!D be a functor, and JX;Y : F (X) ⊗D F (Y ) ! F (X ⊗C Y ) be a natural isomorphism detailing the tensor structure of F . The pair (F; J) is called a quasi-tensor functor if F is exact (i.e., both left exact and right exact), and if F (1C) = 1D. If furthermore (F; J) satisfies the monoidal structure axiom (with the associators in C and D), then (F; J) is just called a tensor functor. There are tons of examples of (quasi)-tensor functors. They include essentially all forgetful functors: indeed, if H ⊂ G is a subgroup, then the canonical forgetful functor Rep(G) ! Rep(H) is tensor. Similarly, if f : H ! G is a group homomorphism, then the 3 pullback functor f ∗ : Rep(G) ! Rep(H) is tensor (the canonical forgetful functor is an example of this when f : H,! G is the inclusion). As we talked about earlier, if F is the !1 !2 identity functor, then there exists no tensor functor F : CG !CG if !1 and !2 are not co- homologous, while if they are cohomologous the tensor functors are classified by H2(G; k×).