Rigidification of algebras over multi-sorted theories

Details Rigidification of algebras over multi-sorted theories Rigidification of algebras over multi-sorted theories

2005-08-08

arxiv.org/abs/math/0508152v3

Algebr. Geom. Topol. 6 (2006) 1925-1955

Mathematics

Algebraic Topology

This is the version published by Algebraic & Geometric Topology on 14 November 2006

Scientific paper

10.2140/agt.2006.6.1925

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.

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