arXiv subjects

Amit Shah

Publications and source records attributed to Amit Shah.

24 records · Page 2Linked to original sources

arXiv2021

Transport of structure in higher homological algebra

We fill a gap in the literature regarding `transport of structure' for (n+2)-angulated, n-exact, n-abelian and n-exangulated categories appearing in (classical and higher) homological algebra. As an application of our main results, we show that a skeleton of one of these kinds of categories inherits the same structure in a canonical way, up to equivalence. In particular, it follows that a skeleton of a weak (n+2)-angulated category is in fact what we call a strong (n+2)-angulated category. When n=1 this clarifies a technical concern with the definition of a cluster category. We also introduce the notion of an n-exangulated functor between n-exangulated categories. This recovers the definition of an (n+2)-angulated functor when the categories concerned are (n+2)-angulated, and the higher analogue of an exact functor when the categories concerned are n-exact.

math.CT↗

arXiv2020

Examples and non-examples of integral categories and the admissible intersection property

Integral categories form a sub-class of pre-abelian categories whose systematic study was initiated by Rump in 2001. In the first part of this article we determine whether several categories of topological and bornological vector spaces are integral. Moreover, we establish that the class of integral categories is not contained in the class of quasi-abelian categories, and that there exist semi-abelian categories that are neither integral nor quasi-abelian. In the last part of the article we show that a category is quasi-abelian if and only if it has admissible intersections, in the sense considered recently by Br{ü}stle, Hassoun and Tattar. This exhibits that a rich class of non-abelian categories having this property arises naturally in functional analysis.

math.CT↗

arXiv2020

Integral and quasi-abelian hearts of twin cotorsion pairs on extriangulated categories

It was shown recently that the heart of a twin cotorsion pair ((S,T),(U,V)) on an extriangulated category is semi-abelian. We provide a sufficient condition for the heart to be integral and another for the heart to be quasi-abelian. This unifies and improves the corresponding results for exact and triangulated categories. Furthermore, if T=U, then we show that the Gabriel-Zisman localisation of the heart at the class of its regular morphisms is equivalent to the heart of the single twin cotorsion pair (S,T). This generalises and improves the known result for triangulated categories, thereby providing new insights in the exact setting.

math.CT↗

arXiv2019

Quasi-abelian hearts of twin cotorsion pairs on triangulated categories

We prove that, under a mild assumption, the heart H of a twin cotorsion pair ((S,T),(U,V)) on a triangulated category C is a quasi-abelian category. If C is also Krull-Schmidt and T=U, we show that the heart of the cotorsion pair (S,T) is equivalent to the Gabriel-Zisman localisation of H at the class of its regular morphisms. In particular, suppose C is a cluster category with a rigid object R and [X_R] the ideal of morphisms factoring through X_R=Ker(Hom(R,-)), then applications of our results show that C/[X_R] is a quasi-abelian category. We also obtain a new proof of an equivalence between the localisation of this category at its class of regular morphisms and a certain subfactor category of C.

math.CT↗

arXiv2019

Auslander-Reiten theory in quasi-abelian and Krull-Schmidt categories

We generalise some of the theory developed for abelian categories in papers of Auslander and Reiten to semi-abelian and quasi-abelian categories. In addition, we generalise some Auslander-Reiten theory results of S. Liu for Krull-Schmidt categories by removing the Hom-finite and indecomposability restrictions. Finally, we give equivalent characterisations of Auslander-Reiten sequences in a quasi-abelian, Krull-Schmidt category.

math.CT↗

arXiv2016

Resource allocation using metaheuristic search

This research is focused on solving problems in the area of software project management using metaheuristic search algorithms and as such is research in the field of search based software engineering. The main aim of this research is to evaluate the performance of different metaheuristic search techniques in resource allocation and scheduling problems that would be typical of software development projects. This paper reports a set of experiments which evaluate the performance of three algorithms, namely simulated annealing, tabu search and genetic algorithms. The experimental results indicate that all of the metaheuristics search techniques can be used to solve problems in resource allocation and scheduling within a software project. Finally, a comparative analysis suggests that overall the genetic algorithm had performed better than simulated annealing and tabu search.

cs.NE↗