Related papers: Framed motivic $\Gamma$-spaces
We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and…
We address some conjectures and open problems in "analysis of symmetries" which include the study of non-commutative harmonic analysis and discontinuous groups for reductive homogeneous spaces beyond the classical framework: (1) discrete…
We study the $\Gamma$-convergence of sequences of free-discontinuity functionals depending on vector-valued functions $u$ which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of…
Enriched motivic $\mathcal A$-spaces are introduced and studied in this paper, where $\mathcal A$ is an additive category of correspondences. They are linear counterparts of motivic $\Gamma$-spaces. It is shown that rational special…
We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel-Voevodsky $\mathbb{A}^1$-homotopy category. As a consequence we show that the motive of a smooth stack (in Voevodsky's triangulated category of…
The machinery of framed (pre)sheaves was developed by Voevodsky [V1]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem [V1] is proved in this paper…
In a number of recent works [6, 7] the authors have introduced and studied a functor $\mathcal{F}_k$ which associates to each loose graph $\Gamma$ -which is similar to a graph, but where edges with $0$ or $1$ vertex are allowed - a…
These notes, written version of a Bourbaki talk, survey Morel-Voevodsky's motivic homotopy theory over a field, with a focus on computations of motivic homotopy sheaves, both stable and unstable. We also describe Isaksen-Wang-Xu's…
By a theorem of Mandell-May-Schwede-Shipley the stable homotopy theory of classical $S^1$-spectra is recovered from orthogonal spectra. In this paper general linear, special linear, symplectic, orthogonal and special orthogonal motivic…
We model equivariant infinite loop spaces indexed on incomplete universes via suitable equivariant analogs of $\Gamma$-spaces. The choice of universe dictates a transfer system which in turn dictates the Segal condition on equivariant…
The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to…
Working over an algebraically closed field of characteristic zero, we compute the cohomology of the subalgebra A(2) of the motivic Steenrod algebra that is generated by Sq^1, Sq^2, and Sq^4. The method of calculation is a motivic version of…
We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group $\Gamma$ with additional cohomological properties. For $\Gamma=\mathbb{Z}_2$ we…
We introduce a general framework realizing edge modes in (classical) gauge field theory as dynamical reference frames, an often suggested interpretation that we make entirely explicit. We focus on a bounded region $M$ with a co-dimension…
A rigidity property for the homotopy invariant stable linear framed presheaves is established. As a consequence a variant of Gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category…
We demonstrate the usefulness of anholonomic frames in the contexts of nonholonomic and vakonomic systems. We take a consistently differential-geometric approach. As an application, we investigate the conditions under which the dynamics of…
In this article we further the study of non-commutative motives. Our main result is the construction of a simple model, given in terms of infinite matrices, for the suspension in the triangulated category of non-commutative motives. As a…
We prove a motivic version of Landweber's exact functor theorem from topology. The main result is that the assignment given by a Landweber-type formula using the MGL-homology of a motivic spectrum defines a homology theory on the stable…
In this article we study in detail the category of noncommutative motives of separable algebras Sep(k) over a base field k. We start by constructing four different models of the full subcategory of commutative separable algebras CSep(k).…
For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers, and finite fields. We use this to extend Morel's identification of the endomorphism ring…