Related papers: K-theory and localizing invariants of large catego…

We compute the value of finitary localizing invariants, including algebraic $K$-theory, on categories of sheaves over stably locally compact spaces $X$. Our formula simultaneously generalizes the cases of locally compact Hausdorff and…

We study a construction of diagrams of dualizable presentable stable $\infty$-categories associated with certain fiber-cofiber sequences over rigid bases, which are sent by localizing invariants, in particular continuous K-theory, to limit…

We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty$-categories) for additive $\infty$-categories, (2) define the notion of nilpotent extensions for…

We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational…

We adapt Grayson's model of higher algebraic $K$-theory using binary acyclic complexes to the setting of stable $\infty$-categories. As an application, we prove that the $K$-theory of stable $\infty$-categories preserves infinite products.

We generalize a recent result of Clausen: For a number field with integers O, we compute the K-theory of locally compact O-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite…

We show that the functor sending a locally compact Hausdorff space $X$ to the $\infty$-category of spectral sheaves $\mathrm{Shv}(X; \mathrm{Sp})$ is initial among all continuous six-functor formalisms on the category of locally compact…

We develop a new framework to study real $K$-theory in the context of $\infty$-categories. For this, we introduce Waldhausen $\infty$-categories with genuine duality, which will be the input for such $K$-theory. These are Waldhausen…

We give a description of the value of a finitary localizing invariant, such as algebraic $K$-theory, on the category of sheaves on a locally coherent space $X$. This in particular includes all spaces that arise as spectra of commutative…

We study the \'etale sheafification of algebraic K-theory, called \'etale K-theory. Our main results show that \'etale K-theory is very close to a noncommutative invariant called Selmer K-theory, which is defined at the level of categories.…

We study algebraic K-theory and topological Hochschild homology in the setting of bimodules over a stable category, a datum we refer to as a laced category. We show that in this setting both K-theory and THH carry universal properties, the…

We introduce the Cuntz-Thomsen picture of $\mathcal{C}$-equivariant Kasparov theory, denoted $\mathrm{KK}^\mathcal{C}$, for a unitary tensor category $\mathcal{C}$ with countably many isomorphism classes of simple objects. We use this…

This paper examines and strengthens the Cuntz-Thomsen picture of equivariant Kasparov theory for arbitrary second-countable locally compact groups, in which elements are given by certain pairs of cocycle representations between C*-dynamical…

We show that for a coconnective ring spectrum satisfying regularity and flatness assumptions, its algebraic K-theory agrees with that of its $\pi_0$. We prove this as a consequence of a more general devissage result for stable infinity…

We present a quick approach to computing the $K$-theory of the category of locally compact modules over any order in a semisimple $\mathbb{Q}$-algebra. We obtain the $K$-theory by first quotienting out the compact modules and subsequently…

The classical trace map is a highly non-trivial map from algebraic K-theory to topological Hochschild homology (or topological cyclic homology) introduced by B\"okstedt, Hsiang and Madsen. It led to many computations of algebraic K-theory…

In this paper we prove some new Stone-type duality theorems for some subcategories of the category $\ZLC$ of locally compact zero-dimensional Hausdorff spaces and continuous maps. These theorems are new even in the compact case. They…

For a finite group $G$, we compute the algebraic $K$-theory of the category of equivariant sheaves on a locally compact Hausdorff $G$-space, generalizing a result of Efimov, and determine the equivariant $E$-theory of the $C^*$-algebra of…

In this paper we study the category of nuclear modules on an affine formal scheme as defined by Clausen and Scholze \cite{CS20}. We also study related constructions in the framework of dualizable and rigid monoidal categories. We prove that…

The present work is the author's doctoral thesis, written during his studies at the University of Bonn. Its goal is to establish the foundations of $K$-theory in the context of adic geometry using the formalism of condensed mathematics and…