Related papers: Brown representability for triangulated categories…
Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a…
We study categories whose objects are the braid representations, i.e. strict monoidal functors $F\colon B\rightarrow Mat$ from the braid category $B$ to the category of matrices $Mat$. Braid representations are equivalent to solutions to…
In this paper we present a new way to construct the pro-category of a category. This new model is very convenient to work with in certain situations. We present a few applications of this new model, the most important of which solves an…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
In this paper we give a purely categorical construction of d-fold matrix factorizations of a natural transformation, for any even integer d. This recovers the classical definition of those for regular elements in commutative rings due to…
The well-known difficulties arising in a classification which is not set-theoretically trivial---involving what is sometimes called a non-smooth quotient---have been overcome in a striking way in the theory of operator algebras by the use…
A representation of $\mathfrak{gl}(V)=V \otimes V^*$ is a linear map $\mu \colon \mathfrak{gl}(V) \otimes M \to M$ satisfying a certain identity. By currying, giving a linear map $\mu$ is equivalent to giving a linear map $a \colon V…
Graduated locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and Boolean algebras. A finitary functor between graduated locally finitely presentable categories is…
Motivated by the study of dependent random variables by coupling with independent blocks of variables, we obtain first sufficient conditions for the moderate deviation principle in its functional form for triangular arrays of independent…
We characterize the generating function of the number of representations described in the title in terms of the theory of modular forms. Appealing to this characterization we obtain explicit formulas for the representation numbers as…
Let $X$ be a smooth projective variety over $\mathbb{C}$ with big (anti-)canonical bundle. It is known that in this situation the Balmer spectrum of the tensor triangulated category of perfect complexes $Perf(X)$ of $X$ equipped with the…
We give a characterization of connected solvable groups in terms of the existence of representations with certain geometric properties. The existence of such representations for the group of upper triangular matrices played an important…
The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of…
Given a thick subcategory of a triangulated category, we define a colocalisation and a natural long exact sequence that involves the original category and its localisation and colocalisation at the subcategory. Similarly, we construct a…
We provide necessary and sufficient conditions for operator-valued functions on arbitrary sets associated with a collection of test functions to have factorizations in several situations.
For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated to a triangle functor from the category on the right to the category on the left. For a morphic…
Working in the setting of ideally exact categories, we investigate the representability of actions of unital non-associative algebras over a field. We show that, in general, such categories fail to be action representable: for instance, the…
Extriangulated categories axiomatize extension-closed subcategories of triangulated categories and generalise both exact categories and triangulated categories. This survey article presents three applications of extriangulated categories to…
In this paper, we are interested in the interplay between integral ternary quadratic forms and class numbers. This is partially motivated by a question of Petersson.
We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the…