Global homotopy theory via partially lax limits

We provide new 1-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 1-category of global spaces is equivalent to a partially lax limit of the functor sending a compact Lie group G to the 1-category of G-spaces.We also prove the stable version of this result, showing that the 1-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 1-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.