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Homotopy theory of presheaves of Gamma-spaces. (English) Zbl 1160.55007

Consider an intermediate model structure on the category of simplicial presheaves over a small Grothendieck site [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 [S. Schwede and B. E. Shipley, Proc. Lond. Math. Soc., III. Ser. 80, No. 2, 491–511 (2000; Zbl 1026.18004)].

MSC:

55P47 Infinite loop spaces
55P42 Stable homotopy theory, spectra
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55P48 Loop space machines and operads in algebraic topology
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