相关论文: Why is Homology so Powerful?
Injectivity of objects with respect to a set $\ch$ of morphisms is an important concept of algebra, model theory and homotopy theory. Here we study the logic of injectivity consequences of $\ch$, by which we understand morphisms $h$ such…
Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision…
For a $C^{*}$-category with a strict $G$-action we construct examples of equivariant coarse homology theories. To this end we first introduce versions of Roe categories of objects in $C^{*}$-categories which are controlled over bornological…
Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…
This survey provides an elementary introduction to operads and to their applications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher…
Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serves as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of…
Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps…
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…
Topological Data Analysis has grown in popularity in recent years as a way to apply tools from algebraic topology to large data sets. One of the main tools in topological data analysis is persistent homology. This paper uses undergraduate…
In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same…
We recently introduced the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. There is a type of idempotent system, said to be symmetric. In the present paper we classify up…
We prove some injectivity theorems. Our proof depends on the theory of mixed Hodge structures on cohomology groups with compact support. Our injectivity theorems would play crucial roles in the minimal model theory for higher-dimensional…
We consider an homogeneous action of a finite group on a free linear category over a field in order to prove that the subcategory of invariants is still free. Moreover we show that the representation type is preserved when considering…
The purpose of this survey article is to introduce the reader to a connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Martin-L\"of into…
We show that the free construction from multicategories to permutative categories is a categorically-enriched non-symmetric multifunctor. Our main result then shows that the induced functor between categories of algebras is an equivalence…
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy…
This paper is a compressed summary of some principal definitions and concepts in the approach to the black box algebra being developed by the authors. We suggest that black box algebra could be useful in cryptanalysis of homomorphic…
Recent advances in our understanding of higher derived limits carry multiple implications in the fields of condensed and pyknotic mathematics, as well as for the study of strong homology. These implications are thematically diverse,…
We compute numerically the homology of several graph complexes in low loop orders, extending previous results.
An algebraic theory $T$ is a category with objects $t_0,t_2...$ such that for each $n$ the object $t_n$ is an $n$-fold categorical product of $t_1$. A strict $T$-algebra is a product preserving functor $A: T\to Spaces$. Lawvere showed that…