Rigidification of homotopy algebras over finite product sketches
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Bruce R. Corrigan-Salter [1]
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[1] Wayne State University
Wayne State University
City of Detroit, Estados Unidos
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- Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 6 (June 2015), 2015, págs. 1962-1991
- Idioma: inglés
- DOI: 10.1016/j.jpaa.2014.07.019
- Enlaces
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Resumen
Multi-sorted algebraic theories provide a formalism for describing various structures on spaces that are of interest in homotopy theory. The results of Badzioch (2002) [1] and Bergner (2006) [4] showed that an interesting feature of this formalism is the following rigidification property. Every multi-sorted algebraic theory defines a category of homotopy algebras, i.e. a category of spaces equipped with certain structure that is to some extent homotopy invariant. Each such homotopy algebra can be replaced by a weakly equivalent strict algebra which is a purely algebraic structure on a space. The equivalence between the homotopy categories of loop spaces and topological groups is a special instance of this result.
In this paper we will introduce the notion of a finite product sketch which is a useful generalization of a multi-sorted algebraic theory. We will show that in the setting of finite product sketches we can still obtain results paralleling these of Badzioch and Bergner, although a rigidification of a homotopy algebra over a finite product sketch is given by a strict algebra over an associated simplicial multi-sorted algebraic theory.
