# Global algebraic K‐theory

@article{Schwede2022GlobalAK, title={Global algebraic K‐theory}, author={Stefan Schwede}, journal={Journal of Topology}, year={2022}, volume={15} }

We introduce a global equivariant refinement of algebraic K‐theory; here ‘global equivariant’ refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global Ω$\Omega$ ‐spectrum that keeps track of genuine G$G$ ‐equivariant infinite loop spaces, for all finite groups G$G$ . The resulting global algebraic K‐theory spectrum is a rigid way of packaging the representation K‐theory, or ‘Swan K‐theory’ into one highly…

## 5 Citations

### G-Global Homotopy Theory and Algebraic K-Theory

- Mathematics
- 2020

We develop the foundations of G-global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand.…

### Parametrised Noncommutative Motives and Equivariant Algebraic K-theory

- Mathematics
- 2022

We construct the parametrised symmetric monoidal category of noncommutative motives associated to the algebraic K-theory of parametrised perfect-stable categories. In the equivariant case for a…

### On the global homotopy theory of symmetric monoidal categories

- Mathematics
- 2020

Parsummable categories were defined by Schwede as the input for his global algebraic $K$-theory construction. We prove that their whole homotopy theory with respect to the so-called global…

### Categorical models of unstable G-global homotopy theory

- Mathematics
- 2021

We prove that the category G-Cat of small categories with Gaction forms a model of unstable G-global homotopy theory for every discrete group G, generalizing Schwede’s global model structure on Cat.…

### Parsummable categories as a strictification of symmetric monoidal categories

- Mathematics
- 2020

We prove that the homotopy theory of parsummable categories (as defined by Schwede) with respect to the underlying equivalences of categories is equivalent to the usual homotopy theory of symmetric…

## References

SHOWING 1-10 OF 41 REFERENCES

### Equivariant algebraic K-theory of G-rings

- Mathematics
- 2015

A group action on the input ring or category induces an action on the algebraic K-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic K-theory is, for example,…

### Global Homotopy Theory

- Mathematics
- 2018

This book introduces a new context for global homotopy theory, i.e., equivariant homotopy theory with universal symmetries. Many important equivariant theories naturally exist not just for a…

### Equivariant algebraic K-theory

- Mathematics
- 1982

There are many ways that group actions enter into algebraic K-theory and there are various theories that fit under the rubric of our title. To anyone familiar with both equivariant topological…

### Infinite Loop G-Spaces Associated to Monoidal G-Graded Categories

- Mathematics
- 1989

We construct a functor KG which takes each pair of monoidal G-graded categories (D,Df) to an infinite loop G-space KG(D,D'). When D'=D, its homotopy groups n%KG(D,D) coincide with the equivariant…

### A convenient setting for equivariant higher algebraic K-theory

- Mathematics
- 1982

^ is a finite group, n the category of finite (left) z-sets, S a q-set, ~ the associated category (see i.i), and Q an exact category in the sense of Quillen [9]. We show that the category [~,Q] of…

### Spectral Mackey functors and equivariant algebraic K-theory, II

- MathematicsTunisian Journal of Mathematics
- 2020

We study the "higher algebra" of spectral Mackey functors, which the first named author introduced in Part I of this paper. In particular, armed with our new theory of symmetric promonoidal…

### Permutative categories, multicategories and algebraicK–theory

- MathematicsAlgebraic & Geometric Topology
- 2009

We show that the $K$-theory construction of arXiv:math/0403403, which preserves multiplicative structure, extends to a symmetric monoidal closed bicomplete source category, with the multiplicative…

### Filtrations of global equivariant K-theory

- Mathematics
- 2015

Arone and Lesh constructed and studied spectrum level filtrations that interpolate between connective (topological or algebraic) K-theory and the Eilenberg-MacLane spectrum for the integers. In this…

### Spectral Mackey functors and equivariant algebraic K-Theory ( I )

- Mathematics
- 2016

Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable∞-category, and we use…