{"title":"Twisting structures and morphisms up to strong homotopy","authors":"Kathryn Hess, Paul-Eugène Parent, Jonathan Scott","doi":"10.1007/s40062-019-00249-w","DOIUrl":"https://doi.org/10.1007/s40062-019-00249-w","url":null,"abstract":"<p>We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the “strong homotopy” morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads <img>, which is exactly the two-sided Koszul resolution of the associative operad <img>, also known as the Alexander-Whitney co-ring.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"15 1","pages":"185 - 222"},"PeriodicalIF":0.5,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00249-w","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4353225","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":"Correction to: Wrong way maps in uniformly finite homology and homology of groups","authors":"Alexander Engel","doi":"10.1007/s40062-019-00246-z","DOIUrl":"https://doi.org/10.1007/s40062-019-00246-z","url":null,"abstract":"","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 4","pages":"1143 - 1144"},"PeriodicalIF":0.5,"publicationDate":"2019-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00246-z","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5142177","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":"A model structure via orbit spaces for equivariant homotopy","authors":"Mehmet Akif Erdal, Aslı Güçlükan İlhan","doi":"10.1007/s40062-019-00241-4","DOIUrl":"https://doi.org/10.1007/s40062-019-00241-4","url":null,"abstract":"<p>Let <i>G</i> be discrete group and <span>(mathcal F)</span> be a collection of subgroups of <i>G</i>. We show that there exists a left induced model structure on the category of right <i>G</i>-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on <i>H</i>-orbits for all <i>H</i> in <span>(mathcal F)</span>. This gives a model categorical criterion for maps that induce weak equivalences on <i>H</i>-orbits to be weak equivalences in the <span>(mathcal F)</span>-model structure.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 4","pages":"1131 - 1141"},"PeriodicalIF":0.5,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00241-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5007558","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":"Cohomology of infinite groups realizing fusion systems","authors":"Muhammed Said Gündoğan, Ergün Yalçın","doi":"10.1007/s40062-019-00240-5","DOIUrl":"https://doi.org/10.1007/s40062-019-00240-5","url":null,"abstract":"<p>Given a fusion system <span>({mathcal {F}})</span> defined on a <i>p</i>-group <i>S</i>, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize <span>({mathcal {F}})</span>. We study these models when <span>({mathcal {F}})</span> is a fusion system of a finite group <i>G</i> and prove a theorem which relates the cohomology of an infinite group model <span>(pi )</span> to the cohomology of the group <i>G</i>. We show that for the groups <i>GL</i>(<i>n</i>,?2), where <span>(nge 5)</span>, the cohomology of the infinite group obtained using the Robinson model is different than the cohomology of the fusion system. We also discuss the signalizer functors <span>(Prightarrow Theta (P))</span> for infinite group models and obtain a long exact sequence for calculating the cohomology of a centric linking system with twisted coefficients.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 4","pages":"1103 - 1130"},"PeriodicalIF":0.5,"publicationDate":"2019-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00240-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4305775","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":"Dense products in fundamental groupoids","authors":"Jeremy Brazas","doi":"10.1007/s40062-019-00238-z","DOIUrl":"https://doi.org/10.1007/s40062-019-00238-z","url":null,"abstract":"<p>Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental group need not imply the well-definedness of countable dense products in the fundamental groupoid. Additionally, we show the fundamental groupoid <span>(Pi _1(X))</span> has well-defined dense products if and only if <i>X</i> admits a generalized universal covering space.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 4","pages":"1083 - 1102"},"PeriodicalIF":0.5,"publicationDate":"2019-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00238-z","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4127323","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":"Minimality in diagrams of simplicial sets","authors":"Carles Broto, Ramón Flores, Carlos Giraldo","doi":"10.1007/s40062-019-00239-y","DOIUrl":"https://doi.org/10.1007/s40062-019-00239-y","url":null,"abstract":"<p>We formulate the concept of minimal fibration in the context of fibrations in the model category <span>({mathbf {S}}^{mathcal {C}})</span> of <span>({mathcal {C}})</span>-diagrams of simplicial sets, for a small index category <span>({mathcal {C}})</span>. When <span>({mathcal {C}})</span> is an <i>EI</i>-category satisfying some mild finiteness restrictions, we show that every fibration of <span>({mathcal {C}})</span>-diagrams admits a well-behaved minimal model. As a consequence, we establish a classification theorem for fibrations in <span>({mathbf {S}}^{mathcal {C}})</span> over a constant diagram, generalizing the classification theorem of Barratt, Gugenheim, and Moore for simplicial fibrations (Barratt?et?al. in Am J Math 81:639–657, 1959).</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 4","pages":"1043 - 1082"},"PeriodicalIF":0.5,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00239-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5160383","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}
Domenico Fiorenza, Fosco Loregian, Giovanni Luca Marchetti
{"title":"Hearts and towers in stable (infty )-categories","authors":"Domenico Fiorenza, Fosco Loregian, Giovanni Luca Marchetti","doi":"10.1007/s40062-019-00237-0","DOIUrl":"https://doi.org/10.1007/s40062-019-00237-0","url":null,"abstract":"<p>We exploit the equivalence between <i>t</i>-structures and normal torsion theories on a stable <span>(infty )</span>-category to show how a few classical topics in the theory of triangulated categories, i.e., the characterization of bounded <i>t</i>-structures in terms of their hearts, their associated cohomology functors, semiorthogonal decompositions, and the theory of tiltings, as well as the more recent notion of Bridgeland’s slicings, are all particular instances of a single construction, namely, the tower of a morphism associated with a <i>J</i>-slicing of a stable <span>(infty )</span>-category <img>, where <i>J</i> is a totally ordered set equipped with a monotone <span>(mathbb {Z})</span>-action.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 4","pages":"993 - 1042"},"PeriodicalIF":0.5,"publicationDate":"2019-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00237-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4875074","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":"Involutions on surfaces","authors":"Daniel Dugger","doi":"10.1007/s40062-019-00236-1","DOIUrl":"https://doi.org/10.1007/s40062-019-00236-1","url":null,"abstract":"<p>We use equivariant surgery to classify all involutions on closed surfaces, up to isomorphism. Work on this problem is classical, dating back to the nineteenth century, with a complete classification finally appearing in the 1990s. In this paper we give a different approach to the classification, using techniques that are more accessible to algebraic topologists as well as a new invariant (which we call the double-Dickson invariant) for distinguishing the “hard” cases.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 4","pages":"919 - 992"},"PeriodicalIF":0.5,"publicationDate":"2019-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00236-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4583710","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":"Comonad cohomology of track categories","authors":"David Blanc, Simona Paoli","doi":"10.1007/s40062-019-00235-2","DOIUrl":"https://doi.org/10.1007/s40062-019-00235-2","url":null,"abstract":"<p>We define a comonad cohomology of track categories, and show that it is related via a long exact sequence to the corresponding <span>(({mathcal {S}}!,!mathcal {O}))</span>-cohomology. Under mild hypotheses, the comonad cohomology coincides, up to reindexing, with the <span>(({mathcal {S}}!,!mathcal {O}))</span>-cohomology, yielding an algebraic description of the latter. We also specialize to the case where the track category is a 2-groupoid.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 4","pages":"881 - 917"},"PeriodicalIF":0.5,"publicationDate":"2019-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00235-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4585662","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":"Matrix factorizations for quantum complete intersections","authors":"Petter Andreas Bergh, Karin Erdmann","doi":"10.1007/s40062-019-00234-3","DOIUrl":"https://doi.org/10.1007/s40062-019-00234-3","url":null,"abstract":"<p>We introduce twisted matrix factorizations for quantum complete intersections of codimension two. For such an algebra, we show that in a given dimension, almost all the indecomposable modules with bounded minimal projective resolutions correspond to such factorizations.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 4","pages":"863 - 880"},"PeriodicalIF":0.5,"publicationDate":"2019-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-019-00234-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5089824","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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