Monoidal categories
type: Item
instance of: scholarly article
title: Monoidal categories
zbMATH Open document ID: 0907.18002
DOI: 10.1007/BF02365308
published in: Journal of Mathematical Sciences (New York)
publication date: 1998-10-22T00:00:00Z
review text: [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 …
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 19D23
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 1214606
zbMATH Keywords: exterior algebra
zbMATH Keywords: lambda ring
zbMATH Keywords: \(K\)-theory of monoidal categories
zbMATH Keywords: tensor categories
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: Amitsur cohomology
zbMATH Keywords: Yang-Baxter operators
zbMATH Keywords: Poincaré series
author: Alexei Davydov
reviewed by: Ross H. Street
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1214606