相关论文: Manifolds, Structures Categorically

The relation between manifold topology, observables and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated to positions and displacements on the manifold. The guiding,…

Let $M$ be a subset of vector space or projective space. The authors define the \emph{generalized configuration space} of $M$ which is formed by $n$-tuples of elements of $M$ where any $k$ elements of each $n$-tuple are linearly…

In this paper we discuss various philosophical aspects of the hyperstructure concept extending networks and higher categories. By this discussion we hope to pave the way for applications and further developments of the mathematical theory…

We present the notion of injective hom-complexity, leading to a connection between the covering number of a group and the sectional number of a group homomorphism, and provide estimates for computing this invariant.

This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often…

We develop a theory of umkehr maps for twisted generalized homology theories. In this theory, interesting umkehr maps, including generalizations of important classical ones, are induced by cartesian morphisms of a certain category opfibred…

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map…

We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a…

The review is devoted to topological global aspects of quantal description. The treatment concentrates on quantizations of kinematical observables --- generalized positions and momenta. A broad class of quantum kinematics is rigorously…

A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets in the category of modules over a commutative algebra is described. Some related structures and (co)homology invariants are discussed, as well as applications to…

We give an elementary construction of polyhedra whose links are connected bipartite graphs, which are not necessarily isomorphic pairwise. We show, that the fundamental groups of some of our polyhedra contain surface groups. In particular,…

The study of complex systems through the lens of category theory consistently proves to be a powerful approach. We propose that cognition deserves the same category-theoretic treatment. We show that by considering a highly-compact cognitive…

Four-manifold theory is employed to study the existence of (twisted) generalized complex structures. It is shown that there exist (twisted) generalized complex structures that have more than one type change loci. In an example-driven…

We define the phrase `category enriched in an fc-multicategory' and explore some examples. An fc-multicategory is a very general kind of 2-dimensional structure, special cases of which are double categories, bicategories, monoidal…

This manuscript develops a geometric approach to ordinary cohomology of smooth manifolds, constructing a cochain complex model based on co-oriented smooth maps from manifolds with corners. Special attention is given to the pull-back product…

Consider the topologically enriched category of compact smooth manifolds (possibly with corners), with morphisms given by codimension zero smooth embeddings. Now formally identify any object X with its thickening X x [-1,1]. We prove that…

Quasirandomness is a general mathematical concept meant to encapsulate several characteristics usually satisfied by random combinatorial objects, and which we regard as describing when a given object 'looks random'. In this survey we…

Despite high-dimensionality of images, the sets of images of 3D objects have long been hypothesized to form low-dimensional manifolds. What is the nature of such manifolds? How do they differ across objects and object classes? Answering…

We introduce a constructive method that provides the local solution of general implicit systems in arbitrary dimension via Hamiltonian type equations. A variant of this approach constructs parametrizations of the manifold, extending the…

We consider the concept of a generalised manifold in the O(d,d) setting, i.e., in double geometry. The conjecture by Hohm and Zwiebach for the form of finite generalised diffeomorphisms is shown to hold. Transition functions on overlaps are…