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Homotopy coherent mapping class group actions and excision for Hochschild complexes of modular categories. (English) Zbl 1473.18008

The authors obtain an extension of the usual Hochschild chain complex of a modular category \(\mathcal{C}\) over an algebraically closed field \(k\) as follows. They show that there is a sequence \((M_g)_{g\geq 0}\) of \(\mathcal{C}\)-bimodules such that the Hochschild chain complex \(CH(\mathcal{C}; M_g)\) of \(\mathcal{C}\) with coefficients in \(M_g\) has a homotopy coherent projective action of the mapping class group of the surface of genus \(g + 1\). The usual Hochschild chain complex of \(\mathcal{C}\) is \(CH(C; M_0)\).

MSC:

18E13 Protomodular categories, semi-abelian categories, Mal’tsev categories
55U35 Abstract and axiomatic homotopy theory in algebraic topology
20F38 Other groups related to topology or analysis
57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
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