Homotopy theory of presheaves of Gamma-spaces (Q1004519)

Consider an intermediate model structure on the category of simplicial presheaves over a small Grothendieck site [\textit{J. F. Jardine}, Can. Math. Bull. 49, No.~3, 407--413 (2006; Zbl 1107.18007)]. This includes the local injective and local projective structures. Then the author constructs on the category of based functors from based finite ordinals to simplicial presheaves (called \(\Gamma\)-spaces) a cofibrantly generated left proper model structure with stable equivalences as weak equivalences. The fibrant objects coincide with the very special \(\Gamma\)-spaces. If the model structure on the category of simplicial presheaves is already monoidal, then the new model structure on \(\Gamma\)-spaces is monoidal as well and satisfies the monoid axiom. Consequently, the category of module objects over a monoid in the category of \(\Gamma\)-spaces and the category of algebra objects over a commutative monoid in the category of \(\Gamma\)-spaces inherit model structures from \(\Gamma\)-spaces by [\textit{S. Schwede} and \textit{B. E. Shipley}, Proc. Lond. Math. Soc., III. Ser. 80, No.~2, 491--511 (2000; Zbl 1026.18004)].