Algebraic & Geometric Topology

Rigidification of algebras over multi-sorted theories

Julia E Bergner

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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.

Article information

Algebr. Geom. Topol., Volume 6, Number 4 (2006), 1925-1955.

Received: 9 August 2005
Revised: 8 September 2006
Accepted: 29 September 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18C10: Theories (e.g. algebraic theories), structure, and semantics [See also 03G30]
Secondary: 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 18E35: Localization of categories 55P48: Loop space machines, operads [See also 18D50]

algebraic theories model categories operads simplicial categories


Bergner, Julia E. Rigidification of algebras over multi-sorted theories. Algebr. Geom. Topol. 6 (2006), no. 4, 1925--1955. doi:10.2140/agt.2006.6.1925.

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