Global homotopy theory via partially lax limits

Linskens, Sil¹ ; Nardin, Denis¹ ; Pol, Luca¹

¹Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany

Geometry & topology, Tome 29 (2025) no. 3, pp. 1345-1440

Résumé

We provide new ∞-categorical models for unstable and stable global homotopy theory. We use the notion of partially lax limits to formalize the idea that a global object is a collection of G-objects, one for each compact Lie group G, which are compatible with the restriction–inflation functors. More precisely, we show that the ∞-category of global spaces is equivalent to a partially lax limit of the functor sending a compact Lie group G to the ∞-category of G-spaces. We also prove the stable version of this result, showing that the ∞-category of global spectra is equivalent to the partially lax limit of a diagram of G-spectra. Finally, the techniques employed in the previous cases allow us to describe the ∞-category of proper G-spectra for a Lie group G, as a limit of a diagram of H-spectra for H running over all compact subgroups of G.

Détail

DOI : 10.2140/gt.2025.29.1345

Keywords: partially lax limits, global homotopy theory, proper equivariant homotopy theory

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