Rigidification of algebras over multi-sorted theories

Bergner, Julia E

doi:10.2140/agt.2006.6.1925

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Rigidification of algebras over multi-sorted theories

Julia E Bergner

Algebraic & Geometric Topology 6 (2006) 1925–1955

DOI: 10.2140/agt.2006.6.1925

arXiv: math.AT/0508152

Abstract

We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different “sorts.” We prove a rigidification result for simplicial algebras over these theories, showing that there is a Quillen equivalence between a model category structure on the category of strict algebras over a multi-sorted theory and an appropriate model category structure on the category of functors from a multi-sorted theory to the category of simplicial sets. In the latter model structure, the fibrant objects are homotopy algebras over that theory. Our two main examples of strict algebras are operads in the category of simplicial sets and simplicial categories with a given set of objects.

Keywords

algebraic theories, model categories, operads, simplicial categories

Mathematical Subject Classification 2000

Primary: 18C10

Secondary: 18G30, 18E35, 55P48

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Publication

Received: 9 August 2005

Revised: 8 September 2006

Accepted: 29 September 2006

Published: 14 November 2006

Authors

Julia E Bergner
Kansas State University 138 Cardwell Hall Manhattan, KS 66506 USA

Volume 6, issue 4 (2006)