  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.
