Related papers: Comparison of Waldhausen constructions
This note is a contribution for a proceedings volume of the workshop "Higher Segal Spaces and their Applications to Algebraic K-Theory, Hall Algebras, and Combinatorics". The content is a streamlined exposition based on a talk about a…
It is known by results of Dyckerhoff-Kapranov and of G\'alvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction has a unital 2-Segal structure. Here, we prove that a certain S.-functor defines an equivalence between the…
In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical…
This note is an expository contribution for a proceedings volume of the workshop "Higher Segal Spaces and their Applications to Algebraic K-theory, Hall Algebras, and Combinatorics". We survey various versions of Waldhausen's S-construction…
We define an $S_\bullet$-construction for squares categories, and introduce a class of squares categories we call "proto-Waldhausen" which capture the properties required for the $S_\bullet$-construction to model the K-theory space. The…
Waldhausen's $S_\bullet$-construction gives a way to define the algebraic $K$-theory space of a category with cofibrations. Specifically, the $K$-theory space of a category with cofibrations $\mathcal{C}$ can be defined as the loop space of…
This is an introduction to Hall algebras from the perspective of $2$-Segal spaces or decomposition spaces, as introduced by Dyckerhoff and Kapranov and G\'{a}lvez-Carrillo, Kock and Tonks, respectively. We explain how linearizations of the…
We give an introduction to the theory of 2-Segal sets, and two of the main applications of them: Hall algebras and a discrete version of Waldhausen's $S_\bullet$-construction. We present several combinatorial examples and how these…
We construct so called Hall monoidal categories (and Hall modules thereover) and exhibit them as a categorification of classical Hall and Hecke algebras (and certain modules thereover). The input of the (functorial!) construction are…
This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to…
We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's…
The theory of 2-Segal sets has connections to various important constructions such as the Waldhausen $S_\bullet$-construction in algebraic $K$-theory, Hall algebras, and (co)operads. In this paper, we construct 2-Segal sets from rooted…
Algebra objects in $\infty$-categories of spans admit a description in terms of $2$-Segal objects. We introduce a notion of span between $2$-Segal objects and extend this correspondence to an equivalence of $\infty$-categories.…
2-Segal spaces arise not only from $S_\dotp$-constructions associated to Waldhausen and (proto) exact categories, but also from $S_\dotp$-constructions associated to certain double-categorical structures. A major step in this direction is…
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 develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable $\infty$-categories. Building on this, we study the relative Waldhausen S-construction $S_\bullet(F)$ of a spherical functor $F$ and…
We introduce higher dimensional analogues of simplicial constructions due to Segal and Waldhausen, respectively producing the direct sum and algebraic $K$-theory spectra of an exact category. We then investigate their fibrancy properties,…
Hall algebras and related constructions have had diverse applications in mathematics and physics, ranging from representation theory and quantum groups to Donaldson-Thomas theory and the algebra of BPS states. The theory of $2$-Segal spaces…
In this paper, we establish a multiplicative equivalence between two multiplicative algebraic $K$-theory constructions, Elmendorf and Mandell's version of Segal's $K$-theory and Blumberg and Mandell's version of Waldhausen's $S_\bullet$…
This is the second part of two papers. In this part, we establish closed formulae for the universal functions in the blowup formulae for virtual Hodge polynomials of Gieseker moduli spaces of rank-2 stable sheaves and Uhlenbeck…