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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 The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories.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
On Singer’s conjecture for the fourth algebraic transfer in certain generic degrees 关于辛格对某些通用度数中第四代数转移的猜想
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-07-13 DOI: 10.1007/s40062-024-00351-8
Ɖặng Võ Phúc
{"title":"On Singer’s conjecture for the fourth algebraic transfer in certain generic degrees","authors":"Ɖặng Võ Phúc","doi":"10.1007/s40062-024-00351-8","DOIUrl":"10.1007/s40062-024-00351-8","url":null,"abstract":"<div><p>Let <i>A</i> be the Steenrod algebra over the finite field <span>(k:= {mathbb {F}}_2)</span> and <i>G</i>(<i>q</i>) be the general linear group of rank <i>q</i> over <i>k</i>. A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, <span>(textrm{Ext}^{q, *}_A(k, k),)</span> for all homological degrees <span>(q geqslant 0.)</span> The Singer algebraic transfer of rank <i>q</i>,  formulated by William Singer in 1989, serves as a valuable method for describing that Ext groups. This transfer maps from the coinvariants of a certain representation of <i>G</i>(<i>q</i>) to <span>(textrm{Ext}^{q, *}_A(k, k).)</span> Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all <span>(qgeqslant 4.)</span> This paper establishes Singer’s conjecture for rank four in the generic degrees <span>(n = 2^{s+t+1} +2^{s+1} - 3)</span> whenever <span>(tne 3)</span> and <span>(sgeqslant 1,)</span> and <span>(n = 2^{s+t} + 2^{s} - 2)</span> whenever <span>(tne 2,, 3,, 4)</span> and <span>(sgeqslant 1.)</span> In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"431 - 473"},"PeriodicalIF":0.7,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141613151","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
Enriched Koszul duality 丰富的科斯祖尔对偶性
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-07-03 DOI: 10.1007/s40062-024-00349-2
Björn Eurenius
{"title":"Enriched Koszul duality","authors":"Björn Eurenius","doi":"10.1007/s40062-024-00349-2","DOIUrl":"10.1007/s40062-024-00349-2","url":null,"abstract":"<div><p>We show that the category of non-counital conilpotent dg-coalgebras and the category of non-unital dg-algebras carry model structures compatible with their closed non-unital monoidal and closed non-unital module category structures respectively. Furthermore, we show that the Quillen equivalence between these two categories extends to a non-unital module category Quillen equivalence, i.e. providing an enriched form of Koszul duality.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"397 - 429"},"PeriodicalIF":0.7,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00349-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508677","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
On the mod 2 cohomology algebra of oriented Grassmannians 论面向格拉斯曼的模 2 同调代数
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-06-28 DOI: 10.1007/s40062-024-00350-9
Milica Jovanović, Branislav I. Prvulović
{"title":"On the mod 2 cohomology algebra of oriented Grassmannians","authors":"Milica Jovanović,&nbsp;Branislav I. Prvulović","doi":"10.1007/s40062-024-00350-9","DOIUrl":"10.1007/s40062-024-00350-9","url":null,"abstract":"<div><p>For <span>(nin {2^t-3,2^t-2,2^t-1})</span> <span>((tge 3))</span> we study the cohomology algebra <span>(H^*(widetilde{G}_{n,3};{mathbb {Z}}_2))</span> of the Grassmann manifold <span>(widetilde{G}_{n,3})</span> of oriented 3-dimensional subspaces of <span>({mathbb {R}}^n.)</span> A complete description of <span>(H^*(widetilde{G}_{n,3};{mathbb {Z}}_2))</span> is given in the cases <span>(n=2^t-3)</span> and <span>(n=2^t-2,)</span> while in the case <span>(n=2^t-1)</span> we obtain a description complete up to a coefficient from <span>({mathbb {Z}}_2.)</span></p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"379 - 396"},"PeriodicalIF":0.7,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529705","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
Two-sided cartesian fibrations of synthetic ((infty ,1))-categories 合成$$(infty ,1)$$-类的双侧笛卡尔纤度
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-06-21 DOI: 10.1007/s40062-024-00348-3
Jonathan Weinberger
{"title":"Two-sided cartesian fibrations of synthetic ((infty ,1))-categories","authors":"Jonathan Weinberger","doi":"10.1007/s40062-024-00348-3","DOIUrl":"10.1007/s40062-024-00348-3","url":null,"abstract":"<div><p>Within the framework of Riehl–Shulman’s synthetic <span>((infty ,1))</span>-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness condition à la Chevalley, Gray, Street, and Riehl–Verity, a two-sided Yoneda Lemma, as well as the proof of several closure properties. Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the two-sided case in a modular fashion. We also briefly discuss <i>discrete</i> two-sided cartesian fibrations in this setting, corresponding to <span>((infty ,1))</span>-distributors. The systematics of our definitions and results closely follows Riehl–Verity’s <span>(infty )</span>-cosmos theory, but formulated internally to Riehl–Shulman’s simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic <span>((infty ,1))</span>-categories correspond to internal <span>((infty ,1))</span>-categories implemented as Rezk objects in an arbitrary given <span>((infty ,1))</span>-topos.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"297 - 378"},"PeriodicalIF":0.7,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00348-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508678","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
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