Bimodules over Operads Characterize Morphisms

Let P be any operad. A P-bimodule R that is a P-cooperad induces a natural “fattening” of the category of P-(co)algebras, expanding the morphism sets while leaving the objects fixed. The morphisms in the resulting R-governed category of P-(co)algebras can be viewed as morphisms “up to R-homotopy” of P-(co)algebras. Let A denote the associative operad in the category of chain complexes. We define a “diffracting” functor Φ that produces A -cooperads 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 -cooperad F that is a minimal, free A -bimodule resolution of A . We show furthermore that the usual bar and cobar constructions are merely “shadows” cast by the diffraction functor. Working with modules over operads further enables us to reformulate the duality of P-algebras and P-coalgebras as chirality of left and right Pmodules, eliminating the need for finite-type conditions.