Related papers: Trace as an alternative decategorification functor
We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with the bimodule category centres of the respective opposite categories and a corresponding categorical equivalence…
We define a cohomology for an arbitrary $K$-linear semistrict semigroupal 2-category $(\mathfrak{C},\otimes)$ (called in the paper a Gray semigroup) and show that its first order (unitary) deformations, up to the suitable notion of…
We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right…
A recently proposed effective action for the trace anomaly describes a tensor-scalar theory that is weakly coupled up to a certain high energy scale, where it becomes strongly interacting. Its ultraviolet completion is obtained by coupling…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the…
In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension…
In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra A are controlled by the second Hochschild cohomology group of A. More generally, any n-parameter first order deformation of A gives, due to…
A notion of Hochschild cohomology of an abelian category was defined by Lowen and Van den Bergh (2005) and they showed the existence of a characteristic morphism from the Hochschild cohomology into the graded centre of the (bounded) derived…
The conjugacy class of a generic unimodular 2 by 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the nineteenth century, it was known that a generic pair (X,Y) of such pairs is determined up to…
Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, for the Lefschetz number of D as the…
Let $G$ be a group and $S$ a unital epsilon-strongly $G$-graded algebra. We construct spectral sequences converging to the Hochschild (co)homology of $S$. Each spectral sequence is expressed in terms of the partial group (co)homology of $G$…
We show that the trace formula interpretation of the explicit formulas expresses the counting function N(q) of the hypothetical curve C associated to the Riemann zeta function, as an intersection number involving the scaling action on the…
A characterization is given for directed graphs that yield graph $C^*$-algebras with continuous trace. This is established for row-finite graphs with no sources first using a groupoid approach, and extended to the general case via the…
A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially…
Hochschild homology is a classical invariant of rings that plays an important role because of its connection to algebraic $K$-theory via the Dennis trace. At level zero, the Dennis trace is induced by the Hattori-Stallings trace. In this…
An n-category is some sort of algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, together with various ways of composing them. We survey various concepts of…
We associate a graded monoidal supercategory $\mathcal{H}\mathit{eis}_{F,k}$ to every graded Frobenius superalgebra $F$ and integer $k$. These categories, which categorify a broad range of lattice Heisenberg algebras, recover many…
It is shown that a recollement of derived categories of algebras induces those of tensor product algebras and opposite algebras respectively, which is applied to clarify the relations between recollements of derived categories of algebras…
Let $\mathcal C$ be a small category with cofibrations. In this paper, we define the $K$-theory and Hochschild homology groups of $\mathcal C$ of order $Y$, where $Y$ is an ordered finite simplicial set with basepoint. Further, we construct…