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Leibniz algebras with derivations 带导数的莱布尼兹代数
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-04-09 DOI: 10.1007/s40062-021-00280-w
Apurba Das
{"title":"Leibniz algebras with derivations","authors":"Apurba Das","doi":"10.1007/s40062-021-00280-w","DOIUrl":"https://doi.org/10.1007/s40062-021-00280-w","url":null,"abstract":"<p>In this paper, we consider Leibniz algebras with derivations. A pair consisting of a Leibniz algebra and a distinguished derivation is called a LeibDer pair. We define a cohomology theory for LeibDer pair with coefficients in a representation. We study central extensions of a LeibDer pair. In the next, we generalize the formal deformation theory to LeibDer pairs in which we deform both the Leibniz bracket and the distinguished derivation. It is governed by the cohomology of LeibDer pair with coefficients in itself. Finally, we consider homotopy derivations on sh Leibniz algebras and 2-derivations on Leibniz 2-algebras. The category of 2-term sh Leibniz algebras with homotopy derivations is equivalent to the category of Leibniz 2-algebras with 2-derivations.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 2","pages":"245 - 274"},"PeriodicalIF":0.5,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00280-w","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4371703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 20
Uniqueness of differential characters and differential K-theory via homological algebra 微分特征的唯一性及同调代数下的微分k理论
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-04-03 DOI: 10.1007/s40062-021-00278-4
Ishan Mata
{"title":"Uniqueness of differential characters and differential K-theory via homological algebra","authors":"Ishan Mata","doi":"10.1007/s40062-021-00278-4","DOIUrl":"https://doi.org/10.1007/s40062-021-00278-4","url":null,"abstract":"<p>Simons and Sullivan constructed a model of differential K-theory, and showed that the differential K-theory functor fits into a hexagon diagram. They asked whether, like the case of differential characters, this hexagon diagram uniquely determines the differential K-theory functor. This article provides a partial affirmative answer to their question: For any fixed compact manifold, the differential K-theory groups are uniquely determined by the Simons–Sullivan diagram up to an isomorphism compatible with the diagonal arrows of the hexagon diagram. We state a necessary and sufficient condition for an affirmative answer to the full question. This approach further yields an alternative proof of a weaker version of Simons and Sullivan’s results concerning axiomatization of differential characters. We further obtain a uniqueness result for generalised differential cohomology groups. The proofs here are based on a recent work of Pawar.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 2","pages":"225 - 243"},"PeriodicalIF":0.5,"publicationDate":"2021-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00278-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4109637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Homotopy theory of monoids and derived localization 一元群的同伦理论及其衍生的局部化
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-03-03 DOI: 10.1007/s40062-021-00276-6
Joe Chuang, Julian Holstein, Andrey Lazarev
{"title":"Homotopy theory of monoids and derived localization","authors":"Joe Chuang,&nbsp;Julian Holstein,&nbsp;Andrey Lazarev","doi":"10.1007/s40062-021-00276-6","DOIUrl":"https://doi.org/10.1007/s40062-021-00276-6","url":null,"abstract":"<p>We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology, both known and new. This includes a recent generalization of Adams’s cobar-construction to the non-simply connected case, and a new algebraic model for the homotopy theory of connected topological spaces as an <span>(infty )</span>-category of discrete monoids.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 2","pages":"175 - 189"},"PeriodicalIF":0.5,"publicationDate":"2021-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00276-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4048577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Homotopical perspective on statistical quantities 统计量的同调透视
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-02-09 DOI: 10.1007/s40062-020-00273-1
Nissim Ranade
{"title":"Homotopical perspective on statistical quantities","authors":"Nissim Ranade","doi":"10.1007/s40062-020-00273-1","DOIUrl":"https://doi.org/10.1007/s40062-020-00273-1","url":null,"abstract":"<p>We introduce the notion of cumulants as applied to linear maps between associative (or commutative) algebras that are not compatible with the algebraic product structure. These cumulants have a close relationship with <span>(A_{infty })</span> and <span>(C_{infty })</span> morphisms, which are the classical homotopical tools for analyzing deformations of algebraically compatible linear maps. We look at these two different perspectives to understand how infinity-morphisms might inform our understanding of cumulants. We show that in the presence of an <span>(A_{infty })</span> or <span>(C_{infty })</span> morphism, the relevant cumulants are strongly homotopic to zero.\u0000</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 1","pages":"155 - 173"},"PeriodicalIF":0.5,"publicationDate":"2021-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-020-00273-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4380422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Smooth functorial field theories from B-fields and D-branes 来自b -场和d -膜的光滑泛函场理论
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-01-23 DOI: 10.1007/s40062-020-00272-2
Severin Bunk, Konrad Waldorf
{"title":"Smooth functorial field theories from B-fields and D-branes","authors":"Severin Bunk,&nbsp;Konrad Waldorf","doi":"10.1007/s40062-020-00272-2","DOIUrl":"https://doi.org/10.1007/s40062-020-00272-2","url":null,"abstract":"<p>In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have boundaries; these are the endpoints of open strings stretched between D-branes. The values of our FFT are obtained from the B-field and its D-branes via transgression. Our construction generalises work of Bunke–Turner–Willerton to include open strings. At the same time, it generalises work of Moore–Segal about open-closed TQFTs to include target spaces. We provide a number of further features of our FFT: we show that it depends functorially on the B-field and the D-branes, we show that it is thin homotopy invariant, and we show that it comes equipped with a positive reflection structure in the sense of Freed–Hopkins. Finally, we describe how our construction is related to the classification of open-closed TQFTs obtained by Lauda–Pfeiffer.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 1","pages":"75 - 153"},"PeriodicalIF":0.5,"publicationDate":"2021-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-020-00272-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4892894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Homotopy types of gauge groups of (mathrm {PU}(p))-bundles over spheres 球上(mathrm {PU}(p)) -束规范群的同伦类型
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-01-21 DOI: 10.1007/s40062-020-00274-0
Simon Rea
{"title":"Homotopy types of gauge groups of (mathrm {PU}(p))-bundles over spheres","authors":"Simon Rea","doi":"10.1007/s40062-020-00274-0","DOIUrl":"https://doi.org/10.1007/s40062-020-00274-0","url":null,"abstract":"<p>We examine the relation between the gauge groups of <span>(mathrm {SU}(n))</span>- and <span>(mathrm {PU}(n))</span>-bundles over <span>(S^{2i})</span>, with <span>(2le ile n)</span>, particularly when <i>n</i> is a prime. As special cases, for <span>(mathrm {PU}(5))</span>-bundles over <span>(S^4)</span>, we show that there is a rational or <i>p</i>-local equivalence <span>(mathcal {G}_{2,k}simeq _{(p)}mathcal {G}_{2,l})</span> for any prime <i>p</i> if, and only if, <span>((120,k)=(120,l))</span>, while for <span>(mathrm {PU}(3))</span>-bundles over <span>(S^6)</span> there is an integral equivalence <span>(mathcal {G}_{3,k}simeq mathcal {G}_{3,l})</span> if, and only if, <span>((120,k)=(120,l))</span>.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 1","pages":"61 - 74"},"PeriodicalIF":0.5,"publicationDate":"2021-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-020-00274-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4819798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
The Segal conjecture for topological Hochschild homology of Ravenel spectra Ravenel谱拓扑Hochschild同调的Segal猜想
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-01-19 DOI: 10.1007/s40062-021-00275-7
Gabriel Angelini-Knoll, J. D. Quigley
{"title":"The Segal conjecture for topological Hochschild homology of Ravenel spectra","authors":"Gabriel Angelini-Knoll,&nbsp;J. D. Quigley","doi":"10.1007/s40062-021-00275-7","DOIUrl":"https://doi.org/10.1007/s40062-021-00275-7","url":null,"abstract":"<p>In the 1980’s, Ravenel introduced sequences of spectra <i>X</i>(<i>n</i>) and <i>T</i>(<i>n</i>) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of <i>X</i>(<i>n</i>), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing the algebraic K-theory of <i>X</i>(<i>n</i>) using trace methods, which approximates the algebraic K-theory of the sphere spectrum in a precise sense. We solve the homotopy limit problem for topological Hochschild homology of <i>T</i>(<i>n</i>) under the assumption that the canonical map <span>(T(n)rightarrow BP)</span> of homotopy commutative ring spectra can be rigidified to map of <span>(E_2)</span> ring spectra. We show that the obstruction to our assumption holding can be described in terms of an explicit class in an Atiyah-Hirzebruch spectral sequence.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 1","pages":"41 - 60"},"PeriodicalIF":0.5,"publicationDate":"2021-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00275-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4750297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
The Adams spectral sequence for 3-local (mathrm {tmf}) 3-local的Adams谱序列 (mathrm {tmf})
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-01-06 DOI: 10.1007/s40062-020-00271-3
D. Culver
{"title":"The Adams spectral sequence for 3-local (mathrm {tmf})","authors":"D. Culver","doi":"10.1007/s40062-020-00271-3","DOIUrl":"https://doi.org/10.1007/s40062-020-00271-3","url":null,"abstract":"<p>The purpose of this article is to record the computation of the homotopy groups of 3-local <span>(mathrm {tmf})</span> via the Adams spectral sequence.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 1","pages":"1 - 40"},"PeriodicalIF":0.5,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-020-00271-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4248332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Groups up to congruence relation and from categorical groups to c-crossed modules 到同余关系的群和从范畴群到c交叉模的群
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2020-11-21 DOI: 10.1007/s40062-020-00270-4
Tamar Datuashvili, Osman Mucuk, Tunçar Şahan
{"title":"Groups up to congruence relation and from categorical groups to c-crossed modules","authors":"Tamar Datuashvili,&nbsp;Osman Mucuk,&nbsp;Tunçar Şahan","doi":"10.1007/s40062-020-00270-4","DOIUrl":"https://doi.org/10.1007/s40062-020-00270-4","url":null,"abstract":"<p>We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to a c-group and to a c-crossed module, which is a connected, special and strict c-crossed module in the sense defined by us. The results obtained here will be applied in the proof of an equivalence of the categories of categorical groups and connected, special and strict c-crossed modules.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"15 3-4","pages":"625 - 640"},"PeriodicalIF":0.5,"publicationDate":"2020-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-020-00270-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4835713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cohomology and deformations of oriented dialgebras 有向对偶的上同调与变形
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2020-09-16 DOI: 10.1007/s40062-020-00265-1
Ali N. A. Koam, Ripan Saha
{"title":"Cohomology and deformations of oriented dialgebras","authors":"Ali N. A. Koam,&nbsp;Ripan Saha","doi":"10.1007/s40062-020-00265-1","DOIUrl":"https://doi.org/10.1007/s40062-020-00265-1","url":null,"abstract":"<p>We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras by mixing the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also introduce a formal deformation theory for oriented dialgebras and show that cohomology of oriented dialgebras controls such deformations.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"15 3-4","pages":"511 - 536"},"PeriodicalIF":0.5,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-020-00265-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4666953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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