Related papers: Gluing Data categories and gluing data functors
We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are…
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
The question "What is category theory" is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of…
We describe a number of geometric contexts where categorification appears naturally: coherent sheaves, constructible sheaves and sheaves of modules over quantizations. In each case, we discuss how "index formulas" allow us to easily perform…
In this paper, we give a geometric construction of string algebras and of their module categories. Our approach uses dissections of punctured Riemann surfaces with extra data at marked points, called labels. As an application, we give a…
A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of…
The aim of this survey is to present applications of covering techniques in the theory of Krull-Gabriel dimension. We start with recalling fundamental facts of the classical covering theory of quivers and locally bounded categories. Then we…
This book gives a thorough introduction to topological data analysis (TDA), the application of algebraic topology to data science. Algebraic topology is traditionally a very specialized field of math, and most mathematicians have never been…
Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is…
We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the…
It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric…
This is a first stab at a mathematical framework in which one can study quantum field theories on spacetimes with quite general geometries. We will study these theories via their factorization algebras. The aim is to identify a minimalist…
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also…
In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp.\…
As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into…
We establish several results on gluing/embedding/extending geometric structures in vacuum spacetimes with a cosmological constant in any spacetime dimensions $d\ge 4$, with emphasis on characteristic data. A useful tool is provided by the…
Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(\mathcal A ,\mathcal…
We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local…
Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched…
We consider several classes of complete intersection numerical semigroups, aris- ing from many different contexts like algebraic geometry, commutative algebra, coding theory and factorization theory. In particular, we determine all the…