A dialogue category is a symmetric monoidal category equipped with a notion of tensorial negation. We establish that the free dialogue category is a category of dialogue games and total innocent strategies. The connection clarifies the algebraic and logical nature of dialogue games, and their intrinsic connection to linear continuations. The proof of the statement is based on an algebraic presentation of dialogue cat- egories inspired by knot theory, and a difficult factorization theorem established by rewriting techniques.