Template games and differential linear logic

We extend our recent template game model of multiplicative additive linear logic (MALL) with an exponential modality of linear logic (LL) derived from the standard categorical construction Sym of the free symmetric monoidal category. We obtain in this way the first game semantics of differential linear logic (DiLL) in its classical form. The construction of the model relies on a careful and healthy comparison with the model of generalised species designed ten years ago by Fiore, Gambino, Hyland and Winskel. Besides the resolution of an old open problem of game semantics, the study reveals an unexpected and promising convergence between linear logic and homotopy theory.