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Dg Loday–Pirashvili modules over Lie algebras 李代数上的Dg Loday-Pirashvili模
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-11-07 DOI: 10.1007/s40062-024-00361-6
Zhuo Chen, Yu Qiao, Maosong Xiang, Tao Zhang
{"title":"Dg Loday–Pirashvili modules over Lie algebras","authors":"Zhuo Chen,&nbsp;Yu Qiao,&nbsp;Maosong Xiang,&nbsp;Tao Zhang","doi":"10.1007/s40062-024-00361-6","DOIUrl":"10.1007/s40062-024-00361-6","url":null,"abstract":"<div><p>A Loday–Pirashvili module over a Lie algebra <span>(mathfrak {g})</span> is a Lie algebra object <span>(bigl (Gxrightarrow {X} mathfrak {g}bigr ))</span> in the category of linear maps, or equivalently, a <span>(mathfrak {g})</span>-module <i>G</i> which admits a <span>(mathfrak {g})</span>-equivariant linear map <span>(X:Grightarrow mathfrak {g})</span>. We study dg Loday–Pirashvili modules over Lie algebras, which is a generalization of Loday–Pirashvili modules in a natural way, and establish several equivalent characterizations of dg Loday–Pirashvili modules. To provide a concise characterization, a dg Loday–Pirashvili module is a non-negative and bounded dg <span>(mathfrak {g})</span>-module <i>V</i> paired with a weak morphism of dg <span>(mathfrak {g})</span>-modules <span>(alpha :Vrightsquigarrow mathfrak {g})</span>. Such a dg Loday–Pirashvili module resolves an arbitrarily specified classical Loday–Pirashvili module in the sense that it exists and is unique (up to homotopy). Dg Loday–Pirashvili modules can also be characterized through dg derivations. This perspective allows the calculation of the corresponding twisted Atiyah classes. By leveraging the Kapranov functor on the dg derivation arising from a dg Loday–Pirashvili module <span>((V,alpha ))</span>, a <span>(hbox {Leibniz}_infty [1])</span> algebra structure can be derived on <span>(wedge ^bullet mathfrak {g}^vee otimes V[1])</span>. The binary bracket of this structure corresponds to the twisted Atiyah cocycle. To exemplify these intricate algebraic structures through specific cases, we utilize this machinery to a particular type of dg Loday–Pirashvili modules stemming from Lie algebra pairs.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 1","pages":"23 - 61"},"PeriodicalIF":0.7,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455566","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
Sequential n-connectedness and infinite deformations of n-loops 序列n-连通性和n-环的无限变形
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-11-04 DOI: 10.1007/s40062-024-00360-7
Jeremy Brazas
{"title":"Sequential n-connectedness and infinite deformations of n-loops","authors":"Jeremy Brazas","doi":"10.1007/s40062-024-00360-7","DOIUrl":"10.1007/s40062-024-00360-7","url":null,"abstract":"<div><p>A space <i>X</i> is “sequentially <i>n</i>-connected” at <span>(xin X)</span> if for every <span>(0leqslant kleqslant n)</span> and sequence of <i>k</i>-loops <span>(f_1,f_2,f_3,ldots :S^krightarrow X)</span> that converges toward the point <i>x</i>, the maps <span>(f_m)</span> contract by a sequence of null-homotopies that converge toward <i>x</i>. Unlike standard local contractibility conditions, the sequential <i>n</i>-connectedness property is closed under forming infinite products and infinite shrinking wedges. We use this property, in conjunction with the Whitney Covering Lemma, to construct homotopies that simultaneously perform infinite deformations of <i>n</i>-loops and, ultimately, allow us to continuously deform arbitrary <i>n</i>-loops into maps with simpler forms. As a direct application, we extend the computation of the <i>n</i>-th homotopy group of a shrinking wedge of certain <span>((n-1))</span>-connected spaces due to K. Eda and K. Kawamura.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 1","pages":"1 - 22"},"PeriodicalIF":0.7,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455463","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
The Hurewicz model structure on simplicial R-modules 简单 R 模块上的胡勒维茨模型结构
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-10-26 DOI: 10.1007/s40062-024-00359-0
Arnaud Ngopnang Ngompé
{"title":"The Hurewicz model structure on simplicial R-modules","authors":"Arnaud Ngopnang Ngompé","doi":"10.1007/s40062-024-00359-0","DOIUrl":"10.1007/s40062-024-00359-0","url":null,"abstract":"<div><p>By a theorem of Christensen and Hovey, the category of non-negatively graded chain complexes has a model structure, called the h-model structure or Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. The Dold–Kan correspondence induces a model structure on the category of simplicial modules. In this paper, we give a description of the two model categories and some of their properties, notably the fact that both are monoidal.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"701 - 723"},"PeriodicalIF":0.7,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587755","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
The (mathbb {Z}/2) Fadell–Husseini index of the complex Grassmann manifolds (G_{n}(mathbb {C}^{2n})) 复格拉斯曼流形 (G_{n}(mathbb {C}^{2n})) 的 Fadell-Husseini 指数
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-10-22 DOI: 10.1007/s40062-024-00357-2
Arijit Nath, Avijit Nath
{"title":"The (mathbb {Z}/2) Fadell–Husseini index of the complex Grassmann manifolds (G_{n}(mathbb {C}^{2n}))","authors":"Arijit Nath,&nbsp;Avijit Nath","doi":"10.1007/s40062-024-00357-2","DOIUrl":"10.1007/s40062-024-00357-2","url":null,"abstract":"<div><p>In this paper, we study the <span>(mathbb {Z}/2)</span> action on complex Grassmann manifolds <span>(G_{n}(mathbb {C}^{2n}))</span> given by taking orthogonal complement. We completely compute the associated <span>(mathbb {Z}/2)</span> Fadell–Husseini index. Our study is parallel to the study of the index of real Grassmann manifolds <span>(G_n(mathbb {R}^{2n}))</span> by Baralić et al. [Forum Math., <b>30</b> (2018), pp. 1539–1572].</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"679 - 700"},"PeriodicalIF":0.7,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587754","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
Flat comodules and contramodules as directed colimits, and cotorsion periodicity 作为有向 colimits 的扁平逗点和反逗点，以及 cotorsion 周期性
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-10-09 DOI: 10.1007/s40062-024-00358-1
Leonid Positselski
{"title":"Flat comodules and contramodules as directed colimits, and cotorsion periodicity","authors":"Leonid Positselski","doi":"10.1007/s40062-024-00358-1","DOIUrl":"10.1007/s40062-024-00358-1","url":null,"abstract":"<div><p>This paper is a follow-up to Positselski and Št’ovíček (Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG]). We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring <span>({mathcal {C}})</span> over a noncommutative ring <i>A</i>, we show that all <i>A</i>-flat <span>({mathcal {C}})</span>-comodules are <span>(aleph _1)</span>-directed colimits of <i>A</i>-countably presentable <i>A</i>-flat <span>({mathcal {C}})</span>-comodules. In the context of a complete, separated topological ring <span>({mathfrak {R}})</span> with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat <span>({mathfrak {R}})</span>-contramodules are <span>(aleph _1)</span>-directed colimits of countably presentable flat <span>({mathfrak {R}})</span>-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of <i>A</i>-flat <span>({mathcal {C}})</span>-comodules and flat <span>({mathfrak {R}})</span>-contramodules as <span>(aleph _1)</span>-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT]). Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion <span>({mathfrak {R}})</span>-contramodules, all the contramodules of cocycles are cotorsion.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"635 - 678"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00358-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lie 2-groups from loop group extensions 来自环群扩展的李2群
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-09-26 DOI: 10.1007/s40062-024-00355-4
Matthias Ludewig, Konrad Waldorf
{"title":"Lie 2-groups from loop group extensions","authors":"Matthias Ludewig,&nbsp;Konrad Waldorf","doi":"10.1007/s40062-024-00355-4","DOIUrl":"10.1007/s40062-024-00355-4","url":null,"abstract":"<div><p>We give a very simple construction of the string 2-group as a strict Fréchet Lie 2-group. The corresponding crossed module is defined using the conjugation action of the loop group on its central extension, which drastically simplifies several constructions previously given in the literature. More generally, we construct strict 2-group extensions for a Lie group from a central extension of its based loop group, under the assumption that this central extension is disjoint commutative. We show in particular that this condition is automatic in the case that the Lie group is semisimple and simply connected.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"597 - 633"},"PeriodicalIF":0.7,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00355-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Transferring algebra structures on complexes 复数上代数结构的转移
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-09-23 DOI: 10.1007/s40062-024-00356-3
Claudia Miller, Hamidreza Rahmati
{"title":"Transferring algebra structures on complexes","authors":"Claudia Miller,&nbsp;Hamidreza Rahmati","doi":"10.1007/s40062-024-00356-3","DOIUrl":"10.1007/s40062-024-00356-3","url":null,"abstract":"<div><p>With the goal of transferring dg algebra structures on complexes along contractions, we introduce a new condition on the associated homotopy, namely a generalized version of the Leibniz rule. We prove that, with this condition, the transfer works to yield a dg algebra (with vanishing descended higher <span>(A_infty )</span> products) and prove that it works also after an application of the Perturbation Lemma even though the new homotopy may no longer satisfy that condition. We also extend these results to the setting of <span>(A_infty )</span> algebras. Then we return to our original motivation from commutative algebra. We apply these methods to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant. The naturality of the construction enables us to find dg algebra homomorphisms between these as well, enabling them to be used as inputs for constructing bar resolutions.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"561 - 596"},"PeriodicalIF":0.7,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00356-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Classification of 2-term (L_infty )-algebras 2 期 $$L_infty $ - 算法的分类
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-09-09 DOI: 10.1007/s40062-024-00354-5
Kevin van Helden
{"title":"Classification of 2-term (L_infty )-algebras","authors":"Kevin van Helden","doi":"10.1007/s40062-024-00354-5","DOIUrl":"10.1007/s40062-024-00354-5","url":null,"abstract":"<div><p>We classify all 2-term <span>(L_infty )</span>-algebras up to isomorphism. We show that such <span>(L_infty )</span>-algebras are classified by a Lie algebra, a vector space, a representation (all up to isomorphism) and a cohomology class of the corresponding Lie algebra cohomology.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"541 - 560"},"PeriodicalIF":0.7,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00354-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The homology digraph of a preordered space 有序空间的同源数图
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-07-27 DOI: 10.1007/s40062-024-00352-7
Catarina Faustino, Thomas Kahl
{"title":"The homology digraph of a preordered space","authors":"Catarina Faustino,&nbsp;Thomas Kahl","doi":"10.1007/s40062-024-00352-7","DOIUrl":"10.1007/s40062-024-00352-7","url":null,"abstract":"<div><p>This paper studies a notion of directed homology for preordered spaces, called the homology digraph. We show that the homology digraph is a directed homotopy invariant and establish variants of the main results of ordinary singular homology theory for the homology digraph. In particular, we prove a Künneth formula, which enables one to compute the homology digraph of a product of preordered spaces from the homology digraphs of the components.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"525 - 540"},"PeriodicalIF":0.7,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00352-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Local systems in diffeology 差异学中的地方系统
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-07-16 DOI: 10.1007/s40062-024-00353-6
Katsuhiko Kuribayashi
{"title":"Local systems in diffeology","authors":"Katsuhiko Kuribayashi","doi":"10.1007/s40062-024-00353-6","DOIUrl":"10.1007/s40062-024-00353-6","url":null,"abstract":"<div><p>By making use of Halperin’s local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by Gómez-Tato, Halperin and Tanré. In the latter half of this article, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with the pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau–de Rham cohomology algebra of the diffeological space. By using the pullback construction, we also discuss a local system model for a topological homotopy pushout.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"475 - 523"},"PeriodicalIF":0.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141643257","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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