2024/01/04 by Linskens, Sil
#Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.02264
We consider the question of cocompleting partially presentable parametrized ∞-categories in the sense of arXiv:2307.11001. As our main result we show that in certain cases one may compute such relative cocompletions via a very explicit formula given in terms of partially lax limits. We then apply this to equivariant homotopy theory, building on the work of op. cit. and arXiv:2301.08240, to conclude that the global ∞-category of globally equivariant spectra is the relative cocompletion of the global ∞-category of equivariant spectra. Evaluating at a group G we obtain a description of the ∞-category of G-global spectra as a partially lax limit, extending the main result of arXiv:2206.01556 for finite groups to G-global homotopy theory. Finally we investigate the question of stabilizing global ∞-categories by inverting the action of representation spheres, and deduce a second universal property for the global ∞-category of globally equivariant spectra, similar to that of arXiv:2302.06207.