  Although the HO/N games are fully abstract for PCF, the traditional notion of
innocence (which underpins these games) is not satisfactory for such language
features as non-determinism and probabilistic branching, in that there are
stateless terms that are not innocent. Based on a category of P-visible plays
with a notion of embedding as morphisms, we propose a natural generalisation by
viewing innocent strategies as sheaves over (a site of) plays, echoing a slogan
of Hirschowitz and Pous. Our approach gives rise to fully complete game models
in each of the three cases of deterministic, nondeterministic and probabilistic
branching. To our knowledge, in the second and third cases, ours are the first
such factorisation-free constructions.
