Applied Categorical Structures

, Volume 24, Issue 6, pp 845–873 | Cite as

A Tale of Three Homotopies

  • Vladimir Dotsenko
  • Norbert Poncin


For a Koszul operad \(\mathcal {P}\), there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy \(\mathcal {P}\)-algebras. Some of those approaches are known to give rise to the same notions. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette.


Homotopy algebras Homotopy morphisms Models for operads 

Mathematics Subject Classifications (2010)

18G55 18D50 


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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of MathematicsTrinity CollegeDublinIreland
  2. 2.Mathematics Research UnitUniversity of LuxembourgLuxembourgGrand Duchy of Luxembourg

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