## Algebraic & Geometric Topology

### Rigidification of algebras over multi-sorted theories

Julia E Bergner

#### 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.

#### Article information

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

Dates
Revised: 8 September 2006
Accepted: 29 September 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796611

Digital Object Identifier
doi:10.2140/agt.2006.6.1925

Mathematical Reviews number (MathSciNet)
MR2263055

Zentralblatt MATH identifier
1125.18003

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796611

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