Monoidal categories
[For the entire collection see Algebra-5 (1998; Zbl 0898.00011).] The intrinsic interest and utility of familiar categorical structures are nicely illustrated in this expository article. Chapter 1 reviews the basic definitions and facts associated with monoidal categories. Unfortunately, different mathematical cultures often use the same terms with different meanings: for example, monoidal functors here are the strong monoidal functors of \textit{S. Eilenberg} and \textit{G. M. Kelly} [``Closed categories'', Proc. Conf. Categor. Algebra, La Jolla 1965, 421-562 (1966; Zbl 0192.10604)] and strong monoidal functors here are usually called strict; quasitensor categories here are braided monoidal categories and tensor categories are symmetric monoidal categories \({\mathcal G}\). For any object \(X\) in such a \(k\)-linear abelian \({\mathcal G}\), each \(k\)-linear representation \((V,\pi)\) of the symmetric group S\(_n\) yields an object \(S(\pi)(X) =V \otimes_{k[\text{S}_n]} X^{\otimes n}\) of \({\mathcal G}\). A particular case (when \(\text{char} k>n)\) is the \(n\)-th exterior power \(\Lambda^nX\). The main construction of the paper occurs in chapter 2. For each monoidal functor \(F:{\mathcal G} \to {\mathcal H}\), there is a natural Hochschild-like cosimplicial monoid \(\text{End} (F^{\otimes*})\) where the elements of dimension \(n\) are natural endomorphisms of the functor \(F^{\otimes n}: {\mathcal G}^n \to{\mathcal H}\) whose value at the object \((X_1, \dots, X_n)\) is the value of \(F\) at \(X_1 \otimes \cdots \otimes X_n\) (bracketed from the left). When dealing with \(k\)-linear categories, the cohomology of the cochain complex associated with the cosimplicial ring \(\text{End} (F^{\otimes*})\) is denoted by \(H^*(F)\). In the case where the rings \(\text{End} (F^{\otimes n})\) are commutative, there is also a cochain complex \(\Aut (F^{\otimes*})\) whose cohomology is denoted by \(h^*(F)\). There is a relationship between alternative (twisted) choices of the constraints \(F(X\otimes Y) \cong F(X) \otimes F(Y)\) and elements of \(h^2(F)\), and, similarly, between alternative choices of the associativity constraints of \({\mathcal G}\) and elements of \(h^3 (\text{Id}_{\mathcal G})\). Chapter 3 discusses aspects of the \(K\)-theory of monoidal categories. \textit{F. Waldhausen} [``Algebraic \(K\)-theory of spaces'', Lect. Notes Math. 1126, 318-419 (1985; Zbl 0579.18006)] defined a pseudofunctor \(K\) from the bicategory of abelian categories, exact functors, and natural transformations to the bicategory of pointed topological spaces, pointed continuous functions, and homotopy classes of homotopies; moreover, \(K\) takes finite products to smash products. The present paper uses this construction to compare \(h^*(F)\) with an actual Hochschild cohomology ring, namely, \(HH^* (K({\mathcal G}), \Omega K({\mathcal H}))\). It is also pointed out that the Grothendieck ring \(K_0 ({\mathcal G})\) of a symmetric monoidal abelian \(k\)-linear category \({\mathcal G}\) becomes a \(\lambda \)-ring under the operations \(\lambda^n ([X])= [\Lambda^nX]\). As a source of examples, categories of modules and comodules over bialgebras are reviewed in chapter 4. The complex \(\text{End} (F^{\otimes*})\) is interpreted in the case where the functor \(F:H \text{-Mod} \to K\text{-Mod}\) is restriction along a bialgebra homomorphism \(f:K\to H\). In particular, when \(H\) is the universal enveloping algebra \(U[{\mathfrak g}]\) of a Lie algebra \({\mathfrak g}\), there is a spectral sequence with \(E_2^{p,q} =H^p ({\mathfrak g}, \Lambda^q {\mathfrak g})\) which converges to \(H^*(F)\) for \(F\) the identity functor of \(U[{\mathfrak g}]\)-Mod; indeed, \(\Lambda^q {\mathfrak g}\) is isomorphic to the cohomology of the forgetful functor \(U[{\mathfrak g}] \text{-Mod} \to\text{Vect}_k\). If \(F\) is the identity functor of the category of bimodules over a commutative \(k\)-algebra \(A\), the cohomology \(h^*(F)\) is the Amitsur cohomology \(H^n(A/k)\) of \(A\), thereby providing a link with the Brauer group of \(A\). Chapter 5 concludes the work by studying the free \(k\)-linear symmetric monoidal category \({\mathcal T}_k\) on a single generator \(X\); it is obtained by Cauchy completing the \(k\)-linearized category of finite sets and bijections. The Grothendieck ring \(K_0 ({\mathcal T}_k)\) is a free \(\lambda\)-ring on a single generator \([X]\). Strong monoidal \(k\)-linear functors \(F:{\mathcal T}_k \to\text{Vect}_k\) correspond to involutory Yang-Baxter operators \(R\) on objects of \(\text{Vect}_k\). Each multiplicative functor \(F:{\mathcal T}_k \to\text{Vect}_k\) (and hence each \(R)\) is ascribed a Poincaré series \[ {\mathcal P}_F (t)= \sum^\infty_{n=0} \dim F( \Lambda^n X) t^n. \] The paper considers the problem of constructing Yang-Baxter operators \(R\) with certain preassigned polynomials as Poincaré series. [There are some problems with the English translation (mainly to do with definite and indefinite articles) and there are some incorrectly numbered references (such as [40] for Proposition 4.4 on page 498 and [11] after Proposition 5.3 on page 500)].
scientific article; zbMATH DE number 1214606
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