Hess, KathrynParent, Paul-EugeneScott, Jonathan2019-11-202019-11-202019-11-20202010.1007/s40062-019-00249-whttps://infoscience.epfl.ch/handle/20.500.14299/163238WOS:000495062300001We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the "strong homotopy" morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring.MathematicsMathematicscomposition productclassifying morphismtwisting cochainkleisli categorystrong homotopy morphismkoszul resolutionalgebraic modelhomologytext::journal::journal article::research article