Global model structures for \(\ast\)-modules (Q2326313)

Global homotopy theory studies equivariant phenomena that occur simultaneously for all compact Lie groups in a compatible way. \textit{S. Schwede} established several Quillen equivalent models for unstable global homotopy theory in [Global homotopy theory. Cambridge: Cambridge University Press (2018; Zbl 1451.55001); Math. Z. 294, No. 1--2, 71--107 (2020; Zbl 1439.55014)], notably the category of orthogonal spaces and the category of \(\mathcal{L}\)-spaces. This paper provides an additional Quillen equivalent model for unstable global homotopy theory, the category of \(\ast\)-modules. The category of \(\ast\)-modules is a full subcategory of the category of \(\mathcal{L} \)-spaces, just as the category of \(S\)-modules in stable homotopy theory is a full subcategory of the category of \(\mathcal{L}\)-spectra. The advantage of the category of \(\ast\)-modules over the whole category of \(\mathcal{L}\)-spaces is that it becomes a symmetric monoidal model category with the box product. Additionally, the author equips the category of monoids in \(\ast\)-modules with a model structure that is Quillen equivalent to Schwede's global model structure on the category of monoids in orthogonal spaces. While there is a model structure on commutative monoids in orthogonal spaces, it is an open question if there exists a Quillen equivalent model structure on commutative monoids in \(\ast\)-modules. The strategy to equip the category of \(\ast\)-modules with a model structure is to identify it with a category of algebras over a monad and check that a variation of a lifting result of [\textit{S. Schwede} and \textit{B. E. Shipley}, Proc. Lond. Math. Soc. (3) 80, No. 2, 491--511 (2000; Zbl 1026.18004)] applies. This approach has previously been used nonequivariantly in [\textit{A. J. Blumberg} et al., Geom. Topol. 14, No. 2, 1165--1242 (2010; Zbl 1219.19006)]. The model structure on the category of monoids in \(\ast\)-modules is established using techniques of [\textit{S. Schwede} and \textit{B. E. Shipley}, Proc. Lond. Math. Soc. (3) 80, No. 2, 491--511 (2000; Zbl 1026.18004)].