Composition of strategies is the crucial operation of game semantics. It corresponds to cut elimination in proof theory. This paper is an attempt to uncover the sheaf-theoretical nature of these two operations. We define a game semantics with a topological flavor for a variant of Multiplicative Additive Linear Logic (henceforth MALL). We show that the standard notion of strategy leads to a correct, yet incomplete model. We then introduce a new, non-standard notion of local'' strategies, which turn out to form a sheaf. <br><br> Composition of strategies is generally divided into two steps: interaction, and hiding. In our setting, interaction arises as gluing in the sheaf of local strategies. Hiding along a cut c: U -> V appears here as an instance of a more general operation,descent'' along c, which also encompasses cut elimination. Descent along c is a morphism of sheaves on V from the direct image along c of local strategies on U, into cut-free local strategies on V. It arises from a factorisation system, roughly dividing plays into their cut-only and cut-free parts.
Finally, our notion of (winning) local strategy is validated by the expected soundness and completeness results w.r.t. MALL provability.