{"title":"Lambda module structure on higher K-groups","authors":"Sourayan Banerjee, Vivek Sadhu","doi":"10.1007/s40062-024-00339-4","DOIUrl":"10.1007/s40062-024-00339-4","url":null,"abstract":"<div><p>In this article, we show that for a quasicompact scheme <i>X</i> and <span>(n>0,)</span> the <i>n</i>-th <i>K</i>-group <span>(K_{n}(X))</span> is a <span>(lambda )</span>-module over a <span>(lambda )</span>-ring <span>(K_{0}(X))</span> in the sense of Hesselholt.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139561241","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}
{"title":"LHS-spectral sequences for regular extensions of categories","authors":"Ergün Yalçın","doi":"10.1007/s40062-024-00338-5","DOIUrl":"10.1007/s40062-024-00338-5","url":null,"abstract":"<div><p>In (Xu, J Pure Appl Algebra 212:2555–2569, 2008), a LHS-spectral sequence for target regular extensions of small categories is constructed. We extend this construction to ext-groups and construct a similar spectral sequence for source regular extensions (with right module coefficients). As a special case of these LHS-spectral sequences, we obtain three different versions of Słomińska’s spectral sequence for the cohomology of regular EI-categories. We show that many well-known spectral sequences related to the homology decompositions of finite groups, centric linking systems, and the orbit category of fusion systems can be obtained as the LHS-spectral sequence of an extension.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139509147","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}
{"title":"Periodic self maps and thick ideals in the stable motivic homotopy category over ({mathbb {C}}) at odd primes","authors":"Sven-Torben Stahn","doi":"10.1007/s40062-023-00337-y","DOIUrl":"10.1007/s40062-023-00337-y","url":null,"abstract":"<div><p>In this article we study thick ideals defined by periodic self maps in the stable motivic homotopy category over <span>({mathbb {C}})</span>. In addition, we extend some results of Ruth Joachimi about the relation between thick ideals defined by motivic Morava K-theories and the preimages of the thick ideals in the stable homotopy category under Betti realization.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138473083","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}
Tanner N. Carawan, Rebecca Field, Bertrand J. Guillou, David Mehrle, Nathaniel J. Stapleton
{"title":"The homotopy of the (KU_G)-local equivariant sphere spectrum","authors":"Tanner N. Carawan, Rebecca Field, Bertrand J. Guillou, David Mehrle, Nathaniel J. Stapleton","doi":"10.1007/s40062-023-00336-z","DOIUrl":"10.1007/s40062-023-00336-z","url":null,"abstract":"<div><p>We compute the homotopy Mackey functors of the <span>(KU_G)</span>-local equivariant sphere spectrum when <i>G</i> is a finite <i>q</i>-group for an odd prime <i>q</i>, building on the degree zero case due to Bonventre and the third and fifth authors.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138473129","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}
{"title":"Prismatic cohomology and p-adic homotopy theory","authors":"Tobias Shin","doi":"10.1007/s40062-023-00335-0","DOIUrl":"10.1007/s40062-023-00335-0","url":null,"abstract":"<div><p>Historically, it was known by the work of Artin and Mazur that the <span>(ell )</span>-adic homotopy type of a smooth complex variety with good reduction mod <i>p</i> can be recovered from the reduction mod <i>p</i>, where <span>(ell )</span> is not <i>p</i>. This short note removes this last constraint, with an observation about the recent theory of prismatic cohomology developed by Bhatt and Scholze. In particular, by applying a functor of Mandell, we see that the étale comparison theorem in the prismatic theory reproduces the <i>p</i>-adic homotopy type for a smooth complex variety with good reduction mod <i>p</i>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136347197","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}
Carmen Constantin, Tobias Fritz, Paolo Perrone, Brandon T. Shapiro
{"title":"Weak cartesian properties of simplicial sets","authors":"Carmen Constantin, Tobias Fritz, Paolo Perrone, Brandon T. Shapiro","doi":"10.1007/s40062-023-00334-1","DOIUrl":"10.1007/s40062-023-00334-1","url":null,"abstract":"<div><p>Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category <span>(Delta )</span> to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in <span>(Delta )</span> to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in <span>(Delta )</span>, which we characterize completely, along with several other classes of squares in <span>(Delta )</span>. Examples of simplicial sets with completeness conditions include quasicategories, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087722","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}
{"title":"On the K-theory of (mathbb {Z})-categories","authors":"Eugenia Ellis, Rafael Parra","doi":"10.1007/s40062-023-00333-2","DOIUrl":"10.1007/s40062-023-00333-2","url":null,"abstract":"<div><p>We establish connections between the concepts of Noetherian, regular coherent, and regular <i>n</i>-coherent categories for <span>(mathbb {Z})</span>-linear categories with finitely many objects and the corresponding notions for unital rings. These connections enable us to obtain a negative <i>K</i>-theory vanishing result, a fundamental theorem, and a homotopy invariance result for the <i>K</i>-theory of <span>(mathbb {Z})</span>-linear categories.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135773684","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}
{"title":"Self-closeness numbers of rational mapping spaces","authors":"Yichen Tong","doi":"10.1007/s40062-023-00332-3","DOIUrl":"10.1007/s40062-023-00332-3","url":null,"abstract":"<div><p>For a closed connected oriented manifold <i>M</i> of dimension 2<i>n</i>, it was proved by Møller and Raussen that the components of the mapping space from <i>M</i> to <span>(S^{2n})</span> have exactly two different rational homotopy types. However, since this result was proved by the algebraic models for the components, it is unclear whether other homotopy invariants distinguish their rational homotopy types or not. The self-closeness number of a connected CW complex is the least integer <i>k</i> such that any of its self-maps inducing an isomorphism in <span>(pi _*)</span> for <span>(*le k)</span> is a homotopy equivalence, and there is no result on the components of mapping spaces so far. For a rational Poincaré complex <i>X</i> of dimension 2<i>n</i> with finite <span>(pi _1)</span>, we completely determine the self-closeness numbers of the rationalized components of the mapping space from <i>X</i> to <span>(S^{2n})</span> by using their Brown–Szczarba models. As a corollary, we show that the self-closeness number does distinguish the rational homotopy types of the components. Since a closed connected oriented manifold is a rational Poincaré complex, our result partially generalizes that of Møller and Raussen.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136208974","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}
{"title":"Comparison of the colimit and the 2-colimit","authors":"Ilia Pirashvili","doi":"10.1007/s40062-023-00331-4","DOIUrl":"10.1007/s40062-023-00331-4","url":null,"abstract":"<div><p>The 2-colimit (also referred to as a pseudo colimit) is the 2-categorical analogue of the colimit and as such, a very important construction. Calculating it is, however, more involved than calculating the colimit. The aim of this paper is to give a condition under which these two constructions coincide. Tough the setting under which our results are applicable is very specific, it is, in fact, fairly important: As shown in a previous paper, the fundamental groupoid can be calculated using the 2-colimit. The results of this paper corresponds precisely to the situation of calculating the fundamental groupoid from a finite covering. We also optimise our condition in the last section, reducing from exponential complexity to a polynomial one.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135591457","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}
{"title":"Localization (C^*-)algebras and index pairing","authors":"Hang Wang, Chaohua Zhang, Dapeng Zhou","doi":"10.1007/s40062-022-00320-z","DOIUrl":"10.1007/s40062-022-00320-z","url":null,"abstract":"<div><p>Kasparov <i>KK</i>-theory for a pair of <span>(C^*)</span>-algebras <span>((A,,B))</span> can be formulated equivalently in terms of the <i>K</i>-theory of Yu’s localization algebra by Dadarlat-Willett-Wu. We investigate the pairings between <i>K</i>-theory <span>(K_j(A))</span> and the two notions of <i>KK</i>-theory which are Kasparov <i>KK</i>-theory <span>(KK_i(A,B))</span> and the localization algebra description of <span>(KK_i(A,B))</span> and show that the two pairings are compatible.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4952614","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}
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