Monoidal categories enriched in braided monoidal categories

Transcription

1 Monoidal categories enriched in braided monoidal categories AMS JMM Special Session on Fusion Categories and Quantum Symmetries David Penneys joint with Scott Morrison January 5, 2017

2 Linear categories - data A linear (Vec enriched) category C has objects a, b, c, C and for every a, b C, an object Vec for every a, b, c C, a composition morphism C(b c) C C(a c) for every a C, an identity morphism id a C(a a), which one can think of as a morphism j a : 1 Vec C(a a).

3 Linear categories - axioms The composition and identity morphisms satisfy the axioms (identity) j a j b. (associativity) C(a d) C(a d). C(b c) C(c d) C(b c) C(c d)

4 V-enriched categories - data Enriched categories were introduced by Eilenberg and Kelly [EK66]. (See also [Kel05].) Let V be a monoidal category. A V-enriched category C has objects a, b, c, C and for every a, b C, an object V for every a, b, c C, a composition morphism C(b c) C C(a c) for every a C, an identity element j a V(1 V C(a a)).

5 V-enriched categories - axioms The composition and identity morphisms satisfy the axioms (identity) j a j b. (associativity) C(a d) C(a d). C(b c) C(c d) C(b c) C(c d)

6 V-monoidal categories To define a V-monoidal category, we require V be braided. Definition [MP17] A (strict) V-monoidal category C is a V-enriched category with a distinguished unit object 1 C C for every a, b C, a tensor product object ab C. for every a, b, c, d C, a tensor product morphism C(a c)c(b d) C C(ab cd) This data satisfies a variety of axioms, the most important being associativity of tensor product and the braided interchange relation.

7 The braided interchange relation Morphisms in an ordinary monoidal category satisfy an exchange relation. If f 1, f 2 C(b c), g 1 C(d e), and g 2 C(e f), we have (f 1 g 1 ) (f 2 g 2 ) (f 1 f 2 ) (g 1 g 2 ). In a V-monoidal category, we replace the ordinary exchange relation with the braided interchange relation: C(ad cf) C(ad cf) C(ad be) C(be cf) C(a c) C(d f) C(d e) C(b c) C(e f) C(d e) C(b c) C(e f)

8 The main theorem Theorem [MP17] Let V be a braided monoidal category. There is a bijective correspondence between: 1. rigid V-monoidal categories C, such that x C(1 C x) admits a left adjoint 2. pairs (T, F Z ) with T a rigid monoidal category and F Z braided oplax monoidal (µ u,v : F Z (uv) F Z (u)f Z (v)) such that F : F Z R admits a right adjoint. V Tr V F Z F Z(T ) These pairs can also be called oplax module tensor categories for V in the spirit of [HPT16a]. T R

9 The underlying tensor category Given a V-monoidal category C, the underlying tensor category C V has the same objects as C, and the hom spaces are given by C V (a b) : V(1 V ). Identity: C(a a) j a Composition: g f C(b c) C(a c) f g C(ac bd) Tensor product: f When C is rigid, so is C V. C(c d) h C f h

10 The categorified trace The functor C(1 C ) : C V V is given by a C(1 C a) and C(1 C b) C V (a b) f C(1 C a) f We only consider (rigid) V-monoidal C such that C(1 C ) has a left adjoint F : V C V. We show that F lifts to a braided oplax monoidal functor F Z : V Z(C V ). In [HPT16a], we showed that when V is braided pivotal, T is pivotal, and F Z : V Z(T ) is pivotal braided strong monoidal, a right adjoint of F F Z R is a categorified trace Tr V : T V.

11 A related result Theorem [HPT16b] Let V be a braided pivotal monoidal category. There is an equivalence of categories between: 1. The category of anchored planar algebras in V The category of pointed module tensor categories for V. These are triples (T, F Z, t) such that T is a pivotal monoidal category F Z : V Z(T ) is a braided pivotal strong monoidal functor such that F F Z R admits a right adjoint t T is a symmetrically self-dual object which generates T as a module tensor category.

12 Example: de-equivariantization Let G be a finite group and T be a rigid monoidal category. Suppose we have a fully faithful strong monoidal functor F Z : Rep(G) Z(T ). Then T /F is a Rep(G)-enriched tensor category. We may apply the braided lax monoidal fiber functor Rep(G) Vec to transport the Rep(G)-enrichment back to Vec. This two step process recovers the usual notion of de-equivariantization. There is a similar quotienting procedure for fiber functors to svec due to [BGH + 16]. This merits further study!

13 De-equivariantization and SU(2) k Consider T SU(2) k, which has fusion graph A k+1. 1 g Let V 1 T, g, which embeds in Z(T ). When k 4n, V Rep(Z/2Z), and we may de-equivariantize to get a D 2(n+1) category. When k 4n + 2, g is a fermion (θ g 1), and V svec. Kevin Walker showed that the case k 4n + 2 gives rise to the D odd spin planar algebras. (See also Jaffe-Liu s para planar algebras [JL16].)

14 Other interesting examples Here are two examples of fully faithful braided strong monoidal functors that we would like to investigate further. Z(Ad(E 8 )) contains a full copy of Fib [BEK01]. 1 τ Z(Ad(4442)) contains a full copy of SU(3) 3 [GI15, Bru16]. W 1 g 2 g

15 Thank you for listening! Monoidal categories enriched in braided monoidal categories. Scott Morrison and David Penneys. Preprint available at arxiv: Planar algebras in braided tensor categories. André Henriques, David Penneys, and James Tener. Preprint available at arxiv:

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