Related papers: Magnitude homology is a derived functor
Magnitude homology is an emerging framework that captures the intrinsic topological and geometric features of metric spaces, demonstrating significant potential for topoplogical data analysis and geometric data analysis. This work…
Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories,…
There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical invariants such as Euler characteristic…
Magnitude homology is an invariant of enriched categories which generalizes ordinary categorical homology -- the homology of the classifying space of a small category. The classifying space can also be generalized in a different direction:…
In this article, we give a framework for studying the Euler characteristic and its categorification of objects across several areas of geometry, topology and combinatorics. That is, the magnitude theory of filtered sets enriched categories.…
Magnitude homology was introduced by Hepworth and Willerton in the case of graphs, and was later extended by Leinster and Shulman to metric spaces and enriched categories. Here we introduce the dual theory, magnitude cohomology, which we…
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…
Consider a diagram of quasi-categories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasi-category admitting…
This work presents an exposition of both the internal structure of derived category of an abelian category D*(A) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented…
We propose a definition of magnitude for a length space with a Borel measure, which involves integrals over the set of geodesics. This quantity agrees with the magnitude of finite metric spaces, up to re-scaling the metric to ensure the…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
In this work we construct an extension for the category of 0-modules by analogy with [H.-J. Baues and G. Wirshing, Cohomology of small categories, J. Pure Appl. Algebra, 38(1985), 187-211]. The 0-cohomology functor becomes a derived functor…
Magnitude is a numerical invariant of metric spaces and graphs, analogous, in a precise sense, to Euler characteristic. Magnitude homology is an algebraic invariant constructed to categorify magnitude. Among the important features of the…
This paper is a review of concepts from graded commutative algebra with specific attention given to length and multiplicity. The author's motivation for this paper comes from the study of equivariant cohomology in algebraic topology where…
Magnitude homology is an $\mathbf{R}^+$-graded homology theory of metric spaces that captures information on the complexity of geodesics. Here we address the question: when are two metric spaces magnitude homology equivalent, in the sense…
A lot of well-known functors such as group homology, cyclic homology of algebras can be described as limits of certain simply defined functors over categories of presentations. In this paper, we develop technique for the description of the…
We study certain integer valued length functions on triangulated categories and establish a correspondence between such functions and cohomological functors taking values in the category of finite length modules over some ring. The…
We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define…
We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of strong…
Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of…