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In this paper, we show all k-linear abelian 1-Calabi-Yau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of…
The relationships between thin elements, commutative shells and connections in cubical omega-categories are explored by a method which does not involve the use of pasting theory or nerves of omega-categories (both of which were previously…
The $C_k$-equivalence is an equivalence relation generated by $C_k$-moves defined by Habiro. Habiro showed that the set of $C_k$-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the…
The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. This extends previous work that appeared in math.QA/0308228. Several important classes of examples of tensor…
It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (CABA's) taking a set to its power-set and, reciprocally, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a…
There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we…
We develop the theory of exact completions of regular $\infty$-categories, and show that the $\infty$-categorical exact completion (resp. hypercompletion) of an abelian category recovers the connective half of its bounded (resp. unbounded)…
If $\Gamma $ is a group, then braided $\Gamma $-crossed modules are classified by braided strict $\Gamma $-graded categorial groups. The Schreier theory obtained for $\Gamma $-module extensions of the type of an abelian $\Gamma $-crossed…
We prove that if two normed-algebra-valued cosine families indexed by a single Abelian group, of which one is bounded and comprised solely of scalar elements of the underlying algebra, differ in norm by less than $1$ uniformly in the…
We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for ($n$-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for…
The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called `algebraic' because they originate from abelian…
We define 2-gerbes bound by complexes of braided group-like stacks. We prove a classification result in terms of hypercohomology groups with values in abelian crossed squares and cones of morphisms of complexes of length 3. We give an…
The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through…
Many types of categorical structure obey the following principle: the natural notion of equivalence is generated, as an equivalence relation, by identifying $A$ with $B$ when there exists a strictly structure-preserving map $A \to B$ that…
In this paper we present a unified proof of the fact that the category of modules over a ring and the category of near-vector spaces in the sense of J. Andr\'e, over an appropriate scalar system (a 'scalar group'), are both abelian…
Building on work of Jasso, we prove that any projectively generated $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category with enough projectives. This supports the claim that $d$-abelian categories…
By the algebraization of affine Nash groups, a connected affine Nash group is an abelian Nash manifold if and only if its algebraization is a real abelian variety. We first classify real abelian varieties up to isomorphisms. Then with a bit…
If X is a CW complex, one can assign to each point of X an ordered abelian group of finite rank whose subset of positive elements depends continuously on the points of X. A locally trivial bundle which arises in this way we denote by E(X).…
For an abelian category $\mathcal{A}$ we investigate when the stable categories $\underline{\mathrm{GPro}}\mathrm{j}(\mathcal{A})$ and $\underline{\mathrm{GIn}}\mathrm{j}(\mathcal{A})$ are triangulated equivalent. To this end, we realize…
Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not…