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By Carl Faith

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Extra resources for Algebra II. Ring Theory: Ring Theory

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By our choice of on decomposable tensors it holds for any F : C ! C 0 and G : D ! D0 and any C D 2 C D: (F 1)(1 G )(C D) = F (C ) G (D) = (1 G )(F 1)(C D) 28 CHAPTER 2. 2-DIMENSIONAL LINEAR ALGEBRA Therefore, we may choose F ;G as the uniquely determined natural isomorphism that is the identity on decomposable functors. The coherence of -;- is obvious, since the identity satis es all coherence laws, whatsoever. Of course F ;G is not altogether the identity. We also determine the braiding and the involutor.

12. De nition: A 2-vector space is a k-linear additive category (C ; ) that admits a subset B C with: Any object X 2 B is simple. For any X 2 B we have dimk (Hom(X; X )) = 1 For any object A 2 C there is a unique nite subset B 0 B, such that A = LXi 2B0 Xini . Any set B C with these properties is called a basis of C . A 2-vector space is called nite, if it admits a nite basis. The 2-category of 2-vector spaces (resp. of nite dimensional 2-vector spaces) is denoted in the sequel with 2-Vec (resp.

If (C ; ) is a cobraided category, then the following are equivalent: 1. (C ; ) is cocommutative. 2. is a monoidal natural transformation. 3. ( ; ; ) and ( ; 0; 0) are cobraided functors. 4. ; 0; ; 0 are comonoidal natural transformations. Proof: Obvious by De nition. 7. 3. BIMONOIDAL AND HOPF CATEGORIES ; ; ; ; D; 0D ; D; 0D ) be bimonoidal categories. A bimonoidal functor is a 5-tuple (F ; ; ; c; c), such that (F ; ; ) is a monoidal functor, (F ; c; c ) is a comonoidal functor and the following four diagrams commute.

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