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Operadic categories as a natural environment for Koszul duality

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 Added by Martin Markl
 Publication date 2018
  fields
and research's language is English




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The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of a large class of operadic categories. In particular, we will prove that operads, in the generalized sense of Batanin-Markl, governing important operad- and/or PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diver
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Batanin and Markls operadic categories are categories in which each map is endowed with a finite collection of abstract fibres -- also objects of the same category -- subject to suitable axioms. We give a reconstruction of the data and axioms of operadic categories in terms of the decalage comonad D on small categories. A simple case involves unary operadic categories -- ones wherein each map has exactly one abstract fibre -- which are exhibited as categories which are, first of all, coalgebras for the comonad D, and, furthermore, algebras for the monad induced on the category of D-coalgebras by the forgetful-cofree adjunction. A similar description is found for general operadic categories arising out of a corresponding analysis that starts from a modified decalage comonad on the arrow category of Cat.
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