2005/05/26 by Kathryn Hess, Hess, Kathryn, Paul-Eugene Parent +3
Mathematics · #18D50 (Primary) #55P30 (Secondary) #57T30 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #math.AT #math.CT #msc:18D50 #msc:55P30 #msc:57T30
paper · pdf · doi:10.48550/arxiv.math/0505559
59 pages
arxiv created 2006/07/10 · arxiv updated 2009/12/01
Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in the category of P-bimodules. The co-ring R induces a natural ``fattening'' of the category of P-(co)algebras, expanding the morphism sets while leaving the objects fixed. Co-rings over operads are thus ``relative operads,'' parametrizing morphisms as operads parametrize (co)algebras. Let A denote the associative operad in the category of chain complexes. We define a ``diffracting'' functor that produces A-co-rings from symmetric sequences of chain coalgebras, leading to a multitude of ``fattened'' categories of (co)associative chain (co)algebras. In particular, we obtain a purely operadic description of the categories DASH and DCSH first defined by Gugenheim and Munkholm, via an A-co-ring that has the two-sided Koszul resolution of A as its underlying A-bimodule. The diffracting functor plays a crucial role in enabling us to prove existence of higher, ``up to homotopy'' structure of morphisms via acyclic models methods. It has already been successfully applied in this sense in a number of recent articles and preprints.