Co-rings over operads characterize morphisms

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dc.creatorHess, Kathryn
dc.creatorParent, Paul-Eugene
dc.creatorScott, Jonathan
dc.date2005-05-26
dc.date2006-07-10
dc.date.accessioned2026-07-07T06:40:03Z
dc.date.available2026-07-07T06:40:03Z
dc.descriptionLet 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.
dc.description59 pages
dc.identifierhttps://arxiv.org/abs/math/0505559
dc.identifierhttp://arxiv.org/abs/math/0505559
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/101301
dc.subjectAlgebraic Topology
dc.subjectCategory Theory
dc.subject18D50 (Primary); 57T30, 55P30 (Secondary)
dc.titleCo-rings over operads characterize morphisms
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