Algebraic & Geometric Topology

Classifying spaces of algebras over a prop

Sinan Yalin

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We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy invariance result which is well known in the case of algebras over operads. The absence of model category structure on algebras over a prop creates difficulties and we introduce new methods to overcome them. We also explain how our result can be extended to algebras over colored props in any symmetric monoidal model category tensored over chain complexes.

Article information

Algebr. Geom. Topol., Volume 14, Number 5 (2014), 2561-2593.

Received: 9 October 2012
Revised: 19 December 2013
Accepted: 24 February 2014
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18G55: Homotopical algebra
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18D50: Operads [See also 55P48]

props classifying spaces moduli spaces bialgebras category homotopical algebra homotopy invariance


Yalin, Sinan. Classifying spaces of algebras over a prop. Algebr. Geom. Topol. 14 (2014), no. 5, 2561--2593. doi:10.2140/agt.2014.14.2561.

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