[Paper Review] Simplicial localization of homotopy algebras over a prop

Sinan Yalin|arXiv (Cornell University)|Jul 18, 2013

Homotopy and Cohomology in Algebraic Topology16 references3 citations

TL;DR

This paper establishes that weak equivalences between cofibrant dg props induce Dwyer-Kan equivalences on the simplicial localizations of their categories of algebras, proving that the homotopy theory of homotopy algebras over a prop is independent of the choice of cofibrant resolution. The result is proven using simplicial localization and complete Segal spaces, extending homotopy invariance to algebras over colored props in combinatorial monoidal dg categories.

ABSTRACT

We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer-Kan equivalence between the simplicial localizations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a prop P does not depend on the choice of a cofibrant resolution of P, and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories.

Motivation & Objective

• To establish homotopy invariance of the homotopy theory of homotopy algebras over a prop under base change via weak equivalences.
• To resolve the coherence problem of whether the homotopy category of homotopy algebras depends on the choice of cofibrant resolution of the underlying prop.
• To extend existing results on operads and props to the broader setting of colored dg props in combinatorial monoidal dg categories.
• To provide a definitive answer to the base change invariance problem in the context of homotopy algebras over props, using simplicial localization techniques.
• To generalize previous results that required strong assumptions (e.g., existence of free algebra functors) to a setting that includes chain complexes over any field.

Proposed method

• Utilizes simplicial localization of relative categories to construct a simplicial category encoding the homotopy theory of algebras over a prop.
• Applies the framework of complete Segal spaces to compare classification diagrams of simplicial localizations.
• Employs the external tensor product and injective model structures on diagram categories to lift homotopical properties from the base category.
• Relies on Theorem 3.14 (from Yalin, 2017) to establish weak equivalence of nerve spaces of quasi-isomorphisms in categories of algebras.
• Applies transitivity of model structures and pushout-product axioms in injective model structures on diagram categories to verify monoidal properties.
• Combines techniques from higher category theory and homotopical algebra to prove Dwyer-Kan equivalence without assuming a model structure on the category of algebras.

Experimental results

Research questions

• RQ1 Does the homotopy category of homotopy algebras over a prop depend on the choice of a cofibrant resolution of the prop?
• RQ2 Can the simplicial localization of the category of algebras over a cofibrant dg prop be made invariant under weak equivalences of the base prop?
• RQ3 Is homotopy invariance under base change for homotopy algebras over props achievable without assuming the existence of a free algebra functor?
• RQ4 Can the homotopy theory of homotopy algebras over a colored prop be shown to be independent of the resolution choice in chain complexes over a field?
• RQ5 How do simplicial localization and complete Segal space techniques interact to establish invariance in the absence of a model structure on algebras?

Key findings

• A weak equivalence between cofibrant dg props induces a Dwyer-Kan equivalence on the simplicial localizations of their respective categories of algebras.
• The homotopy category of homotopy algebras over a prop is independent of the choice of cofibrant resolution, establishing coherence for the notion of algebra up to homotopy.
• The result holds for algebras in combinatorial monoidal dg categories, including chain complexes over any field.
• The proof avoids reliance on model structures on the category of algebras, circumventing a key limitation of prior approaches.
• The method applies to both 1-colored and colored props, with the colored case requiring characteristic zero for full generality.
• Corollary 3.17 confirms that the homotopy theory of homotopy algebras over a colored dg prop over a field of characteristic zero is invariant under base change via weak equivalences of cofibrant resolutions.