Related papers: Correspondences and stable homotopy theory

We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $\Gamma$-objects in 2-categories. In the course of the proof we establish strictfication…

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we…

These notes offer a unified introduction to spectral methods for the study of complex systems. They are intended as an operative manual rather than a theorem-proof textbook: the emphasis is on tools, identities, and perspectives that can be…

In the paper we use the theory of framed correpondences to construct Milnor-Witt transfers on homotopy modules. As a consequence we identify the zeroth stable $\mathbb{A}^1$-homotopy sheaves of smooth varieties with the zeroth homology of…

Arguably, the first bridge between vast, ancient, but disjoint domains of mathematical knowledge, - topology and number theory, - was built only during the last fifty years. This bridge is the theory of spectra in stable homotopy theory.…

As a continuation of our previous work \cite{KV2} the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces…

The study of homotopy theoretic phenomena in the language of type theory is sometimes loosely called `synthetic homotopy theory'. Homotopy theory in type theory is only one of the many aspects of homotopy type theory, which also includes…

We use geometric fixed points to describe the homotopy theory of genuine equivariant commutative ring spectra after inverting the group order. The main innovation is the use of the extra structure provided by the Hill-Hopkins-Ravenel norms…

We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between…

We introduce and study matrix transfers to achieve elementary models for bivariant $K$-theory. They share lots of common properties with Voevodsky's framed correspondences and lead to symmetric matrix motives of algebraic varieties…

This survey contains the main results in rational homotopy, from the beginning to the most recent ones. It makes the status of the art, gives a short presentation of some areas where rational homotopy has been used, and contains a lot of…

The finite stable homotopy category S_0 has been suggested as a candidate for a category of perfect complexes over the monoid scheme Spec F_1. We apply a reconstruction theorem from algebraic geometry to S_0, and show that one recovers the…

Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy-Levine-Panin, a theorem of Cisinski-Deglise and a version of the Roendigs-Ostvaer theorem, rational stable motivic homotopy theory over an infinite perfect…

We show that if G is a finite constant group acting on a scheme X such that the order of G is invertible in the residue fields of X, then the G-equivariant motivic stable homotopy category of X is equivalent to the stabilization of the…

Using a construction closely related to Waldhausen's $S_\bullet$-construction, we produce a spectrum $K(\mathbf{Var}_{/k})$ whose components model the Grothendieck ring of varieties (over a field $k$) $K_0 (\mathbf{Var}_{/k})$. We then…

This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable…

We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of…

We define a symmetric monoidal structure on the parametrised stable homotopy category over a base space with an action of an $E_\infty$ operad. We discuss products, orientations and push-forwards in parametrised cohomology theories…

We establish an isomorphism between the stable homotopy groups of the 2-completed motivic sphere spectrum over the real numbers and the corresponding stable homotopy groups of the 2-completed Z/2-equivariant sphere spectrum, in a certain…

This paper defines homology in homotopy type theory, in the process stable homotopy groups are also defined. Previous research in synthetic homotopy theory is relied on, in particular the definition of cohomology. This work lays the…