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Rigidification of algebras over multi-sorted theories

Julia E Bergner

Algebraic & Geometric Topology 6 (2006) 1925–1955

arXiv: math.AT/0508152


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.

algebraic theories, model categories, operads, simplicial categories
Mathematical Subject Classification 2000
Primary: 18C10
Secondary: 18G30, 18E35, 55P48
Forward citations
Received: 9 August 2005
Revised: 8 September 2006
Accepted: 29 September 2006
Published: 14 November 2006
Julia E Bergner
Kansas State University
138 Cardwell Hall
Manhattan, KS 66506