 

Tobias Lenz
On the global homotopy theory of symmetric monoidal categories
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Published: 
May 13, 2023. 
Keywords: 
Symmetric monoidal categories, parsummable categories,
equivariant algebraic Ktheory, global homotopy theory, Gglobal homotopy theory. 
Subject [2010]: 
55P91 (primary), 19D23 (secondary). 


Abstract
Parsummable categories were introduced by Schwede as input for his global algebraic Ktheory construction. We prove that their whole homotopy theory with respect to the socalled global equivalences can already be modelled by the more mundane symmetric monoidal categories.
In another direction, we show that the resulting homotopy theory is also equivalent to the homotopy theory of a certain simplicial analogue of parsummable categories, that we call parsummable simplicial sets. These form a bridge to several concepts of "globally coherently commutative monoids" like ultracommutative monoids and global Γspaces, that we explore in [3].


Acknowledgements
This article was written as part of my PhD thesis at the University of Bonn, and I would like to thank my advisor Stefan Schwede for suggesting equivariant and global algebraic Ktheory as a thesis topic, as well as for helpful remarks on a previous version of this article. I am moreover grateful to the anonymous referee for various comments that helped improve the exposition of the present article. Finally, I am indebted to Markus Hausmann, who first suggested to me that there should be a notion of Gglobal homotopy theory. This proved to be a fruitful way of thinking about many phenomena surrounding global Ktheory, which in particular permeates [3].
I am grateful to the Max Planck Institute for Mathematics in Bonn for their hospitality and support during my PhD studies. At the time the first version of this article was written, I was an associate member of the Hausdorff Center for Mathematics, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy (GZ 2047/1, project ID 390685813).


Author information
Tobias Lenz
Mathematisches Institut
Rheinische FriedrichWilhelmsUniversität Bonn
Endenicher Allee 60, 53115 Bonn, Germany
and MaxPlanckInstitut für Mathematik
Vivatsgasse 7, 53111 Bonn, Germany
Current address: Mathematical Institute
University of Utrecht
Budapestlaan 6, 3584 CD Utrecht, The Netherlands
t.lenz@uu.nl

