Ernest E. Fontes
We introduce the notion of a bounded weight structure on a stable $\infty$-category and use this to prove the natural generalization of Waldhausen's sphere theorem: We show that the algebraic $K$-theory of a stable $\infty$-category with a bounded non-degenerate weight structure is equivalent to the algebraic $K$-theory of the heart of the weight structure.
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We prove a version of J.P. May's theorem on the additivity of traces, in symmetric monoidal stable $\infty$-categories. Our proof proceeds via a categorification, namely we use the additivity of topological Hochschild homology as an invariant of stable $\infty$-categories and construct a morphism of spectra $\mathrm{THH}(\mathbf C)\to \mathrm{End}(\mathbf 1_\mathbf C)$ for $\mathbf C$ a stably symmetric monoidal rigid $\infty$-category. We also explain how to get a more general statement involving traces of finite (homotopy) colimits.
On the Mac Lane $Q$-Construction for Exact $\infty$-Categories
本文将 McCarthy 的稳定化构造扩展至正合 $\infty$-范畴，通过为任意函子构造相干链复形，直接推广了计算阿贝尔群稳定同调的 Mac Lane 立方 $Q$-复形。
The classical Dold-Kan correspondence is known to admit a categorification in the form of an equivalence between the $\infty$-categories of $2$-simplicial stable $\infty$-categories and connective chain complexes of stable $\infty$-categories. In this work, we extend these concepts to give a categorification of the classical Dwyer-Kan correspondence by showing that the $\infty$-category of $2$-duplicial stable $\infty$-categories is equivalent to the $\infty$-category of conective chain complexes with right adjoints.
Dualizable presentable $\infty$-categories
本文证明可对偶可表示 V-模及其内部左伴随构成的 ∞-范畴本身可表示，并梳理相关形式性质及谱范畴中紧态射的基础性质。
In this paper we define a sequence of monads $\mathbb{T}^(\infty;n)$ $(n\in\mathbb{N})$ on $\infty$-$\mathbb{G}\text{r}$, the category of the $\infty$-graphs. We conjecture that algebras for $\mathbb{T}^(0;n)$ which are defined in a purely algebraic setting, are models of weak $\infty$-groupoids. And for all $n>1$ we conjecture that algebras for $\mathbb{T}^(\infty;n)$ which are defined in a purely algebraic setting, are models of weak $(\infty; n)$-categories.
Dévissage for Algebraic K-theory of Small Stable $\infty$-categories
本文扩展了heart定理，将其推广到一般的小稳定∞-范畴，建立了精确函子诱导非负K-群同构的充要条件，该条件满足分解条件。
Keller sequences of exact $\infty$-categories
该研究推广稳定∞-范畴的Verdier序列为正合∞-范畴的Keller序列，证明代数K-理论对其局部化，据此给出Quillen分解定理的范畴论证明，并利用Keller的映射空间刻画推导Keller定理的另一证明。
For every regular cardinal $α$, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially DG categories which are stable under suspensions, cosuspensions, cones and $α$-small sums. Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.
We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty$-categories) for additive $\infty$-categories, (2) define the notion of nilpotent extensions for suitable $\infty$-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas-Goodwillie-McCarthy theorem for stable $\infty$-categories which are not monogenically generated (such as the stable $\infty$-category of Voevodsky's motives or the stable $\infty$-category of perfect complexes on s…
Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. This work provides a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed pres…
For a full subcategory B of a unital A_infinity-category C a quotient unital A_infinity-category `C/B' is defined. For differential graded categories such quotient is constructed by V.Drinfeld. Our construction is explicit and uses freely generated A_infinity-categories. The quotient D=`C/B' represents the A_infinity-2-functor A \mapsto A_\infty^u(C,A)_{modulo B}, which associates with a given unital A_infinity-category A the A_infinity-category of unital A_infinity-functors C -> A, whose restriction to B is contractible.
Algebraic K-theory of stable $\infty$-categories via binary complexes
本文通过二元复形方法，在稳定∞-范畴中构建了代数K理论，并证明其保持无限乘积性质。
Controlled objects in left-exact $\infty$-categories and the Novikov conjecture
本研究构造了带G作用的左正合∞-范畴的等变X受控对象范畴，结合代数K-理论得到新等变粗同调理论，将相关装配映射单射性结果应用于左正合∞-范畴的代数K-理论。
$ A_\infty $-Deformations and their Derived Categories
本文研究带曲率$A_\infty$-形变的导出范畴定义难题，通过迭代优化曲率与同调分裂，建立形变Kadeishvili定理，并用树构造极小模型上的$A_\infty$-结构。
Algebraic $K_0$ for unpointed homotopy Categories
本文为无基点 Waldhausen 范畴和稳定 $\infty$-范畴引入 Grothendieck 堆概念，成功将代数 $K_0$ 研究扩展至无基点拓扑空间的同伦范畴。
We prove a twisted Bass-Heller-Swan decomposition for both the connective and the non-connective K-theory spectrum of additive categories.
We prove that algebraic G-theory in is representable in unstable and stable motivic homotopy categories; in the stable category we identify it with the Borel-Moore theory associated to algebraic K-theory, and show that such an identification is compatible with the functorialities defined by Quillen and Thomason.
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