Special Issue "Homological and Homotopical Algebra and Category Theory"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 March 2016).

Special Issue Editors

Prof. Dr. Hvedri Inassaridze
Website
Guest Editor
A. Razmadze Mathematical Institute of Tbilisi State University, 6, Tamarashvili Str., Tbilisi 0177, Georgia
Interests: K-theory; homotopical algebra; category theory; non-commutaive geometry
Prof. Dr. Antonio Cegarra

Guest Editor
Department of Algebra Faculty of Sciences, University of Granada, 18071 Granada, Spain
Interests: category theory; homological and homotopical algebra
Prof. Dr. Marino Gran

Guest Editor
Institut de Recherche en Mathématique et Physique Université Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgique
Interests: categorical algebra; Galois theory; universal algebra
Prof. Dr. Michael Batanin

Guest Editor
Macquarie University, Sydney, NSW, Australia
Interests: operads and their generalisations; Hochschild cohomology; enriched categories; higher categories; model categories
Prof. Dr. Zurab Janelidze

Guest Editor
Department of Mathematical Sciences, Stellenbosch University, South Africa
Interests: categorical algebra

Special Issue Information

Dear Colleagues,

The foundations of homological and homotopical algebra go back to the latter half of the previous century, to the works of S. Eilenberg, S. MacLane, and H. Cartan, and further to follow A. Dold and D. Quillen. Its aim is to study the properties of algebraic objects (groups, associative rings and algebras, Lie algebras, modules, monoids) using the methods of algebraic topology. It can also be viewed as a linearized version of homotopy theory of homotopy types. Nowadays, homological and homotopical algebra is a profound branch of mathematics, a fundamental and essential tool useful for many areas of mathematics, for example, class field theory, algebraic topology, and homotopy theory. That is illustrated by the famous Serre’s Conjecture, regarding the relationship between projective modules and free modules over polynomial rings. This problem of homological and commutative algebra, coming from algebraic geometry, was affirmatively proven in 1976. It has also helped to rewrite the foundations of algebraic geometry, to prove Weil Conjectures, and to create very powerful areas, such as homology theory of groups, Hochschild and cyclic homology theories, and algebraic K-theory. It should be noted that K-theory (algebraic and topological), based on fundamental works of A. Grothendieck, M. Atiyah, H. Bass, J. Milnor, and D.Quillen, is closely related to homotopical algebra and non-commutative geometry having applications to theoretical physics (string theory).

The beginnings of category theory go to A. Grothendieck and F.W. Lawvere in the 20th century. Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science. It is a general mathematical theory formalizing mathematical structures and their concept in terms of objects and arrows called morphisms. It is a powerful language, allowing us to see the universal components of a family of structures of a given kind, and how mathematical structures of different kinds are interrelated. It can be applied to mathematical logic and it is an alternative to set theory as a foundation of mathematics. Categorical methods are successfully used in homological algebra and  algebraic topology.

Prof. Dr. Hvedri Inassaridze
Prof. Dr. Antonio Cegarra
Prof. Dr. Marino Gran
Prof. Dr. Michael Batanin
Prof. Dr. Zurab Janelidze
Guest Editors

Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed Open Access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. For this issue the Article Processing Charge (APC) will be waived for well-prepared manuscripts. English correction and/or formatting fees of 250 CHF (Swiss Francs) will be charged in certain cases for those articles accepted for publication that require extensive additional formatting and/or English corrections.

Keywords

  • Chain complexes and simplicial objects
  • Commutative algebra
  • Cohomology of groups
  • Algebraic K-theory
  • Homotopy theory
  • Derived functors
  • Homology theory of associative rings
  • Spectral sequences
  • Higher categories
  • Cyclic homology
  • Derived category
  • Additive and homotopy functors
  • Categorical algebra
  • Projective dimension of rings

Published Papers (3 papers)

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Research

Open AccessArticle
Elimination of Quotients in Various Localisations of Premodels into Models
Mathematics 2017, 5(3), 37; https://doi.org/10.3390/math5030037 - 09 Jul 2017
Abstract
The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/Ω-spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a [...] Read more.
The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/ Ω -spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allow one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an elimination of quotients and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak ( ω , n ) -categories, weak ω -groupoids and higher moduli stacks. Full article
(This article belongs to the Special Issue Homological and Homotopical Algebra and Category Theory)
Open AccessArticle
Cohen Macaulayness and Arithmetical Rank of Generalized Theta Graphs
Mathematics 2016, 4(3), 43; https://doi.org/10.3390/math4030043 - 29 Jun 2016
Abstract
In this paper, we study some algebraic invariants of the edge ideal of generalized theta graphs, such as arithmetical rank, big height and height. We give an upper bound for the difference between the arithmetical rank and big height. Moreover, all Cohen-Macaulay (and [...] Read more.
In this paper, we study some algebraic invariants of the edge ideal of generalized theta graphs, such as arithmetical rank, big height and height. We give an upper bound for the difference between the arithmetical rank and big height. Moreover, all Cohen-Macaulay (and unmixed) graphs of this type will be characterized. Full article
(This article belongs to the Special Issue Homological and Homotopical Algebra and Category Theory)
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Open AccessArticle
A Cohomology Theory for Commutative Monoids
Mathematics 2015, 3(4), 1001-1031; https://doi.org/10.3390/math3041001 - 27 Oct 2015
Cited by 1
Abstract
Extending Eilenberg–Mac Lane’s cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for [...] Read more.
Extending Eilenberg–Mac Lane’s cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for the three-cohomology classes in terms of braided monoidal groupoids. Full article
(This article belongs to the Special Issue Homological and Homotopical Algebra and Category Theory)
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