Related papers: On the Q construction for exact quasicategories

Utilizing simplicial Waldhausen theory, we prove that the geometric realization of the topologized category of bounded chain complexes over complex numbers (resp. real numbers) is an infinite loop space that represents connective complex…

We prove a bicategorical analogue of Quillen's Theorem A. As an application, we deduce the well-known result that a pseudofunctor is a biequivalence if and only if it is essentially surjective on objects, essentially full on 1-cells, and…

We introduce a bicategory that refines the localization of the category of dg categories with respect to quasi-equivalences and investigate its properties via formal category theory. Concretely, we first introduce the bicategory of dg…

The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $ G $ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group.…

We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category O. We prove an…

We propose a new notion of unbounded $K\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $\sigma$-unital $C^{*}$-algebras, we can then associate a semigroup…

We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient…

The theory of support varieties gives a rich supply of examples of thick subcategories of the stable module category of a finite group algebra. We study direct sum decompositions of such categories. We give examples where there are finer…

For any ring $A$ and a small, preadditive, Hom-finite, and locally bounded category $Q$ that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors from $Q$ to the category of…

In the literature, there are two standard rank filtrations on $K$-theory: an ``unstable'' one which is traditionally defined through the homology of $GL_n$, and a ``stable'' one which was defined by Rognes using the simplicial structure on…

Let $X$ be a simplicial set. We construct a novel adjunction between the categories of retractive spaces over $X$ and of $X_{+}$-comodules, then apply recent work on left-induced model category structures (arXiv:1401.3651v2…

We algorithmically compute integral Eilenberg-MacLane homology of all semigroups of order at most $8$ and present some particular semigroups with notable classifying spaces, refuting conjectures of Nico. Along the way, we give an…

We extend Mazzola's counterpoint model using category theory, generalizing from the category $\mathbf{Set}$ to other topoi with suitable properties. This generalization suggests that counterpoint's essential structure depends on specific…

The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented…

We associate to every $G$-bornological coarse space $X$ and every left-exact $\infty$-category with $G$-action a left-exact infinity-category of equivariant $X$-controlled objects. Postcomposing with algebraic K-theory leads to {new}…

We construct a notion of derived completion which applies to homomorphisms of commutative S-algebras. We study the relationship of the construction with other constructions of completions, and prove various invariance properties. The…

We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C*-algebras agrees…

We show that the quasicategory defined as the localization of the category of (simple) graphs at the class of A-homotopy equivalences does not admit colimits. In particular, we settle in the negative the question of whether the A-homotopy…

We quiver-interpret the classical simplicial theory - including the cosimplex category $\Delta$, Dold-Kan correspondence, and Hochschild homology - as a certain Q-homotopy theory of type $A$. For the cyclic and cubical theories, we proceed…

We construct a new infinite-dimensional family of homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms of the two-sphere. Moreover, for any constant $K$ less than the total area of the sphere, we produce unbounded…