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A cochain level proof of Adem relations in the mod 2 Steenrod algebra mod2 Steenrod代数中Adem关系的协链水平证明
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-08-19 DOI: 10.1007/s40062-021-00287-3
Greg Brumfiel, Anibal Medina-Mardones, John Morgan
{"title":"A cochain level proof of Adem relations in the mod 2 Steenrod algebra","authors":"Greg Brumfiel,&nbsp;Anibal Medina-Mardones,&nbsp;John Morgan","doi":"10.1007/s40062-021-00287-3","DOIUrl":"10.1007/s40062-021-00287-3","url":null,"abstract":"<div><p>In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-<i>i</i> products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod <i>p</i> operations for all primes <i>p</i>. Steenrod’s student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod’s original cochain definition of the Square operations.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00287-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5039495","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: 9
Relative singularity categories and singular equivalences 相对奇异范畴和奇异等价
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-08-18 DOI: 10.1007/s40062-021-00289-1
Rasool Hafezi
{"title":"Relative singularity categories and singular equivalences","authors":"Rasool Hafezi","doi":"10.1007/s40062-021-00289-1","DOIUrl":"10.1007/s40062-021-00289-1","url":null,"abstract":"<div><p>Let <i>R</i> be a right noetherian ring. We introduce the concept of relative singularity category <span>(Delta _{mathcal {X} }(R))</span> of <i>R</i> with respect to a contravariantly finite subcategory <span>(mathcal {X} )</span> of <span>({text {{mod{-}}}}R.)</span> Along with some finiteness conditions on <span>(mathcal {X} )</span>, we prove that <span>(Delta _{mathcal {X} }(R))</span> is triangle equivalent to a subcategory of the homotopy category <span>(mathbb {K} _mathrm{{ac}}(mathcal {X} ))</span> of exact complexes over <span>(mathcal {X} )</span>. As an application, a new description of the classical singularity category <span>(mathbb {D} _mathrm{{sg}}(R))</span> is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00289-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4710235","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 equivalence between Feynman transform and Verdier duality 费曼变换与维迪尔对偶的等价性
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-07-23 DOI: 10.1007/s40062-021-00286-4
Hao Yu
{"title":"The equivalence between Feynman transform and Verdier duality","authors":"Hao Yu","doi":"10.1007/s40062-021-00286-4","DOIUrl":"10.1007/s40062-021-00286-4","url":null,"abstract":"<div><p>The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality to twisted modular operads and the Feynman transform. Specifically, for each twisted modular operad <span>(mathcal {P})</span> (taking values in dg-vector spaces over a field <i>k</i> of characteristic 0), there is a certain sheaf <span>(mathcal {F})</span> associated with it on the moduli space of stable metric graphs such that the Verdier dual sheaf <span>(Dmathcal {F})</span> is associated with the Feynman transform <span>(Fmathcal {P})</span> of <span>(mathcal {P})</span>. In the course of the proof, we also prove a relation between cyclic operads and modular operads originally proposed in the pioneering work of Getzler and Kapranov; however, to the best knowledge of the author, no proof has appeared. This geometric interpretation in operad theory is of fundamental importance. We believe this result will illuminate many aspects of the theory of modular operads and find many applications in the future. We illustrate an application of this result, giving another proof on the homotopy properties of the Feynman transform, which is quite intuitive and simpler than the original proof.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00286-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4893099","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 the K(1)-local homotopy of (mathrm {tmf}wedge mathrm {tmf}) 的K(1)-局部同伦 (mathrm {tmf}wedge mathrm {tmf})
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-07-20 DOI: 10.1007/s40062-021-00283-7
Dominic Leon Culver, Paul VanKoughnett
{"title":"On the K(1)-local homotopy of (mathrm {tmf}wedge mathrm {tmf})","authors":"Dominic Leon Culver,&nbsp;Paul VanKoughnett","doi":"10.1007/s40062-021-00283-7","DOIUrl":"10.1007/s40062-021-00283-7","url":null,"abstract":"<div><p>As a step towards understanding the <span>(mathrm {tmf})</span>-based Adams spectral sequence, we compute the <i>K</i>(1)-local homotopy of <span>(mathrm {tmf}wedge mathrm {tmf})</span>, using a small presentation of <span>(L_{K(1)}mathrm {tmf})</span> due to Hopkins. We also describe the <i>K</i>(1)-local <span>(mathrm {tmf})</span>-based Adams spectral sequence.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00283-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4792783","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
2-Segal objects and algebras in spans 跨度中的2-分段对象和代数
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-05-17 DOI: 10.1007/s40062-021-00282-8
Walker H. Stern
{"title":"2-Segal objects and algebras in spans","authors":"Walker H. Stern","doi":"10.1007/s40062-021-00282-8","DOIUrl":"https://doi.org/10.1007/s40062-021-00282-8","url":null,"abstract":"<p>We define a category parameterizing Calabi–Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2-Segal cyclic objects in C to Calabi–Yau algebra objects in Span(C).</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00282-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4694359","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: 6
Torsion in the magnitude homology of graphs 图的大小同调中的扭转
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-05-15 DOI: 10.1007/s40062-021-00281-9
Radmila Sazdanovic, Victor Summers
{"title":"Torsion in the magnitude homology of graphs","authors":"Radmila Sazdanovic,&nbsp;Victor Summers","doi":"10.1007/s40062-021-00281-9","DOIUrl":"https://doi.org/10.1007/s40062-021-00281-9","url":null,"abstract":"<p>Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We analyze the structure and implications of torsion in magnitude homology. We show that any finitely generated abelian group may appear as a subgroup of the magnitude homology of a graph, and, in particular, that torsion of a given prime order can appear in the magnitude homology of a graph and that there are infinitely many such graphs. Finally, we provide complete computations of magnitude homology of a class of outerplanar graphs and focus on the ranks of the groups along the main diagonal of magnitude homology.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00281-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4620860","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: 8
Derived categories of NDG categories NDG类别的衍生类别
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-03-31 DOI: 10.1007/s40062-021-00279-3
Jun-ichi Miyachi, Hiroshi Nagase
{"title":"Derived categories of NDG categories","authors":"Jun-ichi Miyachi,&nbsp;Hiroshi Nagase","doi":"10.1007/s40062-021-00279-3","DOIUrl":"https://doi.org/10.1007/s40062-021-00279-3","url":null,"abstract":"<p>In this paper we study N-differential graded categories and their derived categories. First, we introduce modules over an N-differential graded category. Then we show that they form a Frobenius category and that its homotopy category is triangulated. Second, we study the properties of its derived category and give triangle equivalences of Morita type between derived categories of N-differential graded categories. Finally, we show that this derived category is triangle equivalent to the derived category of some ordinary differential graded category.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00279-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5182032","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 the relative K-group in the ETNC, Part II 论etc中的相对k群，第二部分
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2020-11-06 DOI: 10.1007/s40062-020-00267-z
Oliver Braunling
{"title":"On the relative K-group in the ETNC, Part II","authors":"Oliver Braunling","doi":"10.1007/s40062-020-00267-z","DOIUrl":"https://doi.org/10.1007/s40062-020-00267-z","url":null,"abstract":"<p>In a previous paper we showed that, under some assumptions, the relative <i>K</i>-group in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a <i>K</i>-group of locally compact equivariant modules. Our approach as well as the standard one both involve presentations: One due to Bass–Swan, applied to categories of finitely generated projective modules; and one due to Nenashev, applied to our topological modules without finite generation assumptions. In this paper we provide an explicit isomorphism.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-020-00267-z","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4269379","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
Homotopy Gerstenhaber algebras are strongly homotopy commutative 同伦Gerstenhaber代数是强同伦可交换的
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2020-11-01 DOI: 10.1007/s40062-020-00268-y
Matthias Franz
{"title":"Homotopy Gerstenhaber algebras are strongly homotopy commutative","authors":"Matthias Franz","doi":"10.1007/s40062-020-00268-y","DOIUrl":"https://doi.org/10.1007/s40062-020-00268-y","url":null,"abstract":"<p>We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff–Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a <span>(mathbin {cup _1})</span>-product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense of Munkholm.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-020-00268-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4051819","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: 6
Homotopic distance between functors 函子间的同伦距离
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2020-10-13 DOI: 10.1007/s40062-020-00269-x
E. Macías-Virgós, D. Mosquera-Lois
{"title":"Homotopic distance between functors","authors":"E. Macías-Virgós,&nbsp;D. Mosquera-Lois","doi":"10.1007/s40062-020-00269-x","DOIUrl":"https://doi.org/10.1007/s40062-020-00269-x","url":null,"abstract":"<p>We introduce a notion of <i>categorical homotopic distance between functors</i> by adapting the notion of homotopic distance in topological spaces, recently defined by the authors, to the context of small categories. Moreover, this notion generalizes the work on categorical LS-category of small categories by Tanaka.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-020-00269-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4557847","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: 6
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