UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Wednesday, 1 June, 2016 1:30 pm to 4:30 pm PAPER 122 TOPICS IN CATEGORY THEORY Attempt no more than ONE question from Section I and TWO questions from Section II. There are SIX questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 SECTION I 1 (a) Give a description of the free braided monoidal category Fbr on the terminal category 1, and a description of the free braided strict monoidal category B on the terminal category 1. (You are not requested to prove the respective universal properties. ) (b) Prove that there is a canonical braided strict monoidal functor Fbr > B that is an equivalence. (You may use the coherence theorem for monoidal categories. ) 2 (a) Define the notions of opmonoidal functor and opmonoidal natural transformation. (b) Define opmonodial monad. Show that, if a monad on a monoidal category is opmonoidal, then its Eilenberg-Moore category of algebras carries a monoidal structure, and that the associated forgetful functor is strict monoidal. (c) Deduce that if H is a bialgebra over a commutative ring k, prove that the category the left H-modules is monoidal and the associated forgetful functor into k-Mod is strict monoidal. (d) Continuing with the previous part, prove that if H is Hopf, then the category of left H-modules is monoidal left closed. Part III, Paper 122
3 UNIVERSITY OF CAMBRIDGE 3 SECTION II Let V and W be two monoidal categories. A Frobenius monoidal functor V > W is a functor F: V > W equipped with a monoidal structure (F, 40, 4) and an opmonoidal structure (F, vo, v) that satisfy the following axioms. FX & F(Y &Z)-FX & (FY & FZ). a-1 HI (FX & FY) ® FZ Ix, YOZ Ix,Y®1 (1) F(X ® (Y &Z)) Fa 1 LF((X &Y) & Z) UXIY,Z |F(X & Y ) & FZ F(X ØY) OFZ ~- /(FX & FY) & FZ ? FX ® (FY & FZ) Ux,Y®1 OXIY,Z F((X & Y) ® Z) F& # F(X ® (Y & Z) -FX & F(Y ® Z) 10ly,z (2) (a) Suppose that e: XY -> I and n: I -> Y & X are the evaluation and coevaluation of a dual pair in V. Prove that a Frobenius structure on the functor F: V > W as above makes 40, I DO FI En > F(YOX) UY.X > FYRFX. FX®FY 0x,Y F(XOY) Fe> FI I the evaluation and coevaluation of a dual pair. Let j: I -> A + A & A: m be a monoid in the monoidal category V. A coseparable structure is a morphism &: A -> I such that E . m is the evaluation of a dual pair (making A dual to itself). We say that (A, j, m, E) is