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Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture

Published online by Cambridge University Press:  01 December 2004

Joachim Kock  and
Bertrand Toën
Joachim Kock
Affiliation:
Laboratoire de Mathematiques, J. A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, Francetoen@math.unice.fr
Bertrand Toën
Affiliation:
Laboratoire de Mathematiques, J. A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, Francetoen@math.unice.fr
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Abstract

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We show that if $(M,\otimes,I)$ is a monoidal model category then $\mathbb{R}\underline{\rm End}_{M}(I)$ is a (weak) 2-monoid in sSet. This applies in particular when M is the category of A-bimodules over a simplicial monoid A: the derived endomorphisms of A then form its Hochschild cohomology, which therefore becomes a simplicial 2-monoid.

Keywords

model categoriessimplicial categoriesDeligne conjectureHochschild cohomology18G55 (primary)18D05 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 141 , Issue 1 , January 2005 , pp. 253 - 261
Copyright
Foundation Compositio Mathematica 2005