Exact triangles, Koszul duality, and coisotopic boundary conditions

We develop a theory of “arrowed” (operads and) dioperads, which are to exact triangles as dioperads are to vector spaces. A central example to this paper is the arrowed operad controlling “derived ideals” for any operad. The Koszul duality theory of arrowed dioperads interacts well with rotation of exact triangles, and in particular with “exact Stars of David,” which are pairs of exact triangles drawn on top of each other in an interesting way. Using this framework, we give a cochain-level lift of the “relative Poincar'e duality” enjoyed by oriented manifolds with boundary; moreover, our cochain-level lift satisfies a natural locality-type condition, and is uniquely determined by this property. We discuss the meaning of the words “relative orientation” and “coisotropic.” We extend the AKSZ construction to bulk-boundary settings with Poisson bulk fields and coisotropic boundary conditions.