Representation theory of Hopf Categories by Martin Neuchl

By Martin Neuchl

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In a way this 2-category is to take the place, the category of vector spaces has, if one deals with ordinary Hopf algebras. We x once and for all an algebraically closed eld k. We denote by vec, (resp. Vec) the category of nite dimensional k-vector spaces (resp. the category of all k-vector spaces). From now on, Ck always denotes the 2-category of small k-linear additive categories, with the k-linear functors as 1-morphisms. By k-linear we mean that the categories are enriched in vec. This 2-category admits a Gray monoid structure and a strongly involutory 2-braiding that we develop as a main goal of this chapter.

Yi )i2f1;:::;ng) an arbitrary family of morphisms in a k-linear, additive category, fi : Xi ! Yi is a monomorphism, if and only if fi is for all i. We conclude X i(U ) = 0 for i 6= j and therefore U = Xj ^ X j(U ). Since U is a subobject of Xj , we have U = Xj . 18. Lemma: Any idempotent matrix with nonnegative integer entries is either the unit matrix, or has a zero line or a zero column. Proof: Let M = (ai;j ) be an idempotent matrix with nonnegative integer entries. Being a projection, M has a positive trace, unless M = 0.

Vec. We now want to study some properties of such objects. 18. De nition: Let (C ; ; ; c; c; c) be a comonoidal category. A cogebra is a weak comonoidal functor (C; ; ) : vec ! C . A comonoidal natural transformation f : (C; ; ) ) (D; 0; 0) between two cogebras is called a cogebra morphism. 4. COGEBRAS AND REPRESENTATIONS OF BIGEBRAS 49 Hence, a cogebra consists of an object C 2 C , a morphism : A A ! A(1) A(2) in C C and a morphism : k ! (A) in vec, such that the following diagrams commute. 1 A A A A(1) 1 ?

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