Related papers: A 2-dimensional torsion theory on symmetric monoid…

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We…

We introduce the notion of a bicollapsible 2-complex. This allows us to generalize the hyperbolicity of one-relator groups with torsion to a broader class of groups with presentations whose relators are proper powers. We also prove that…

Previous work (Pradines, 1966, Aof and Brown, 1992) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This…

We develop a homotopy theory for additive categories endowed with endofunctors, analogous to the concept of a model structure. We use it to construct the homotopy theory of a Hovey triple (which consists of two compatible complete cotorsion…

Building on structure observed in equivariant homotopy theory, we define an equivariant generalization of a symmetric monoidal category: a $G$-symmetric monoidal category. These record not only the symmetric monoidal products but also…

This paper considers non-Abelian homology groups of a group diagram introduced as homotopy groups of a simplicial change. We prove a theorem stating that the non-Abelian homology groups of a group diagram are isomorphic to the homotopy…

First, we show that a compact object $C$ in a triangulated category, which satisfies suitable conditions, induces a $t$-structure. Second, in an abelian category we show that a complex $P^{\centerdot}$ of small projective objects of term…

In this article, we interconnect two different aspects of higher category theory, in one hand the theory of infinity categories and on an other hand the theory of 2-categories.We construct an explicit functorial path objet in the model…

Inspired by a remarkable work of F\'{e}lix, Halperin and Thomas on the asymptotic estimation of the ranks of rational homotopy groups, and more recent works of Wu and the authors on local hyperbolicity, we prove two asymptotic formulae for…

The Hecke algebras for all symmetric groups taken together form a braided monoidal category that controls all quantum link invariants of type A and, by extension, the standard canon of topological quantum field theories in dimension 3 and…

We extend the notion of rational points and cohomological obstructions on varieties to categories fibred in groupoids. We also establish the generalized theory of descent by torsors. Then we interpret the obstruction given by the second…

The main purpose of this paper is to introduce the structure of soft group category. In this category, we determine some special objects and morphisms having a universal structure such as the final object and product. Therefore, the…

In this paper is presented a new approach to the axiomatic homotopy theory in categories, which offers a simpler and more useful answer to this old question: how two objects in a category (without any topological feature) can be deformed…

A toroidal group is a generalization of a complex torus, and is obtained as the quotient of the complex Euclidean space $\mathbb{C}^n$ by a discrete subgroup. Toroidal groups with finite-dimensional cohomology, called theta toroidal groups,…

The notion of a $(\varphi,\hat{G})$-module is defined by Tong Liu in 2010 to classify lattices in semi-stable representations. In this paper, we study torsion $(\varphi,\hat{G})$-modules, and torsion p-adic representations associated with…

We describe torsion classes in the first cohomology group of $\text{SL}_2(\mathbb{Z})$. In particular, we obtain generalized Dickson's invariants for p-power polynomial rings. Secondly, we describe torsion classes in the zero-th homology…

We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically using up-to-homotopy operad…

The purpose of this article is to prove that the category of cocommutative Hopf $K$-algebras, over a field $K$ of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it…

We consider commutative Frobenius pseudomonoids in the bicategory of spans, and we show that they are in correspondence with 2-Segal cosymmetric sets. Such a structure can be interpreted as a coherent 2-dimensional topological quantum field…

In this paper we discuss generalizations of discrete torsion to noninvertible symmetries in 2d QFTs. One point of this paper is to explain that there are two complementary generalizations. Both generalizations are counted by $H^2(G,U(1))$…