Related papers: Efimov K-theory and universal localizing invariant

We apply the notion of relative adjoint functor to generalise closed monoidal categories. We define representations in such categories and give their relation with left actions of monoids. The translation of these representations under lax…

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. Using this framework, we then introduce…

We construct a symmetric spectrum representing the G-equivariant K-theory of C*-algebras for a compact group or a proper groupoid G. Our spectrum is functorial for equivariant *-homomorphisms. We use this to establish the additivity of the…

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth K-oriented…

We introduce a notion of Milnor square of stable $\infty$-categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision…

In frames of the nonlocal and nonpolynomial quantum theory of the one component scalar field in $D$-dimensional spacetime, stated by Gariy Vladimirovich Efimov, the expansion of the $\mathcal{S}$-matrix is revisited for different…

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…

We show that Mandell's inverse $K$-theory functor from $\Gamma$-categories to permutative categories preserves multiplicative structure. This is a first step towards an equivariant generalization that would be inverse to the construction of…

We construct Kasparov's bifunctor $KK$ and $E$-theory by stable homotopy theoretic methods. This is motivated by results concerning constructions of bivariant theories on more general categories such as, for example, bornological algebras.…

Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of…

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…

This survey article on bivariant Kasparov theory and E-theory is mainly intended for readers with a background in homotopical algebra and category theory. We approach both bivariant K-theories via their universal properties and equip them…

We give a general formula for the equivariant complex $K$-theory $K_G^*(V)$ of a finite dimensional real linear space $V$ equipped with a linear action of a compact group $G$ in terms of the representation theory of a certain double cover…

We compute the trace of an endomorphism in equivariant bivariant K-theory for a compact group G in several ways: geometrically using geometric correspondences, algebraically using localisation, and as a Hattori-Stallings trace. This results…

We first retell in the K-theoretic context the heuristics of $S^1$-equivariant Floer theory on loop spaces which gives rise to $D_q$-module structures, and in the case of toric manifolds, vector bundles, or super-bundles to their explicit…

In this paper, we construct an equivariant coarse homology theory with values in the category of non-commutative motives of Blumberg, Gepner and Tabuada, with coefficients in any small additive category. Equivariant coarse K-theory is…

The Fourier coefficients F(t) of a function f on a compact symmetric space U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter t, which…

This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov $K$-theory spectra of $k$-algebras. These are shown to be homotopy invariant, excisive in each variable $K$-theories. We prove that the spectra…

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 introduce the most general to date version of the permutation-equivariant quantum K-theory, and express its total descendant potential in terms of cohomological Gromov-Witten invariants. This is the higher-genus analogue of adelic…