Related papers: Iterated magnitude homology
Multimodal normal incestual systems are investigated in terms of multiple categories. The different sorted composition of operators are exhibited as 2-cells in multiple categories built up from 2-categories giving rise to different axioms.…
We investigate the homological behaviour of compactly generated triangulated categories under separable extensions. We show that homological invariants (finiteness of global dimension, gorensteinness and regularity) are preserved under such…
We introduce the notion of iterated group extensions, which, roughly speaking, is what one obtains by forming a group extension of a group extension. We interpret iterated extensions in terms of group cohomology, in the same way as…
In this dissertation, we compare the "classical" homology of an $\omega$-category (defined as the homology of its Street nerve) with its polygraphic homology. More precisely, we prove that both homologies generally do not coincide and call…
The recently introduced notions of guarded traced (monoidal) category and guarded (pre-)iterative monad aim at unifying different instances of partial iteration whilst keeping in touch with the established theory of total iteration and…
We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories…
For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We…
Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary for invertible structures to have…
We compute the magnitude (an isometric invariant of metric spaces) of compact $\mathbb{R}$-trees and show that it equals $1 + L/2$, where $L \in [0, \infty]$ denotes the total length. Although length is the only geometric invariant captured…
In this paper, a novel parallel hybrid iterative method is proposed for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of variational inequalities for inverse strongly monotone…
The magnitude of a finite metric space has recently emerged as a novel invariant quantity, allowing to measure the effective size of a metric space. Despite encouraging first results demonstrating the descriptive abilities of the magnitude,…
Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the metric varies. Motivated by applications in topological data…
We construct an iterative method for factorising small strict n-categories into a unique (up to isomorphism) collection of small 1- categories. Following this we develop the theory to include a large class of $\infty$-categories. We use…
We characterize injective objects, injective hulls and essential embeddings in the category of symmetric categories enriched in a small, integral and involutive quantaloid. In particular, injective partial metric spaces are precisely…
The familiar adjunction between ordered sets and completely distributive lattices can be extended to generalised metric spaces, that is, categories enriched over a quantale (a lattice of "truth values"), via an appropriate distributive law…
In this paper we explore the connection between the ranks of the magnitude homology groups of a graph and the structure of its subgraphs. To this end, we introduce variants of magnitude homology called eulerian magnitude homology and…
We introduce the theory of enrichment over an internal monoidal category as a common generalization of both the standard theories of enriched and internal categories. The aim of the paper is to justify and contextualize the new notion by…
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…
We introduce a new metric homology theory, Moderately Discontinuous Homology, which captures Lipschitz properties of metric subanalytic germs. The main novelty is to allow "moderately discontinuous" chains, which are specially advantageous…
We consider a class of non-locally compact groups on which one may define a left-invariant, finitely additive measure taking values in some finitely generated extension of the field $\mathbb{R}$ of real numbers. In particular, we recover…