New York Journal of Mathematics
Volume 29 (2023), 635-686


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 K-theory, global homotopy theory, G-global homotopy theory.
Subject [2010]: 55P91 (primary), 19D23 (secondary).

Parsummable categories were introduced by Schwede as input for his global algebraic K-theory construction. We prove that their whole homotopy theory with respect to the so-called 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 ultra-commutative monoids and global Γ-spaces, that we explore in [3].


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 K-theory 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 G-global homotopy theory. This proved to be a fruitful way of thinking about many phenomena surrounding global K-theory, 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 Friedrich-Wilhelms-Universität Bonn
Endenicher Allee 60, 53115 Bonn, Germany
and Max-Planck-Institut für Mathematik
Vivatsgasse 7, 53111 Bonn, Germany

Current address: Mathematical Institute
University of Utrecht
Budapestlaan 6, 3584 CD Utrecht, The Netherlands