Co-rings over Operads Characterize Morphisms

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 Φ 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 F , which 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 [13], [11], [9], [12] and [14].