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A theorem on multiplicative cell attachments with an application to Ravenel’s X(n) spectra 乘法细胞附着物的一个定理及其在Ravenel X(n)谱中的应用
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-11-15 DOI: 10.1007/s40062-018-0222-6
Jonathan Beardsley
{"title":"A theorem on multiplicative cell attachments with an application to Ravenel’s X(n) spectra","authors":"Jonathan Beardsley","doi":"10.1007/s40062-018-0222-6","DOIUrl":"https://doi.org/10.1007/s40062-018-0222-6","url":null,"abstract":"<p>We show that the homotopy groups of a connective <span>(mathbb {E}_k)</span>-ring spectrum with an <span>(mathbb {E}_k)</span>-cell attached along a class <span>(alpha )</span> in degree <i>n</i> are isomorphic to the homotopy groups of the cofiber of the self-map associated to <span>(alpha )</span> through degree 2<i>n</i>. Using this, we prove that the <span>(2n-1)</span>st homotopy groups of Ravenel’s <i>X</i>(<i>n</i>) spectra are cyclic for all <i>n</i>. This further implies that, after localizing at a prime, <span>(X(n+1))</span> is homotopically unique as the <span>(mathbb {E}_1-X(n))</span>-algebra with homotopy groups in degree <span>(2n-1)</span> killed by an <span>(mathbb {E}_1)</span>-cell. Lastly, we prove analogous theorems for a sequence of <span>(mathbb {E}_k)</span>-ring Thom spectra, for each odd <i>k</i>, which are formally similar to Ravenel’s <i>X</i>(<i>n</i>) spectra and whose colimit is also <i>MU</i>.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 3","pages":"611 - 624"},"PeriodicalIF":0.5,"publicationDate":"2018-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0222-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4623345","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: 4
Isotropic reductive groups over discrete Hodge algebras 离散Hodge代数上的各向同性约化群
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-11-10 DOI: 10.1007/s40062-018-0221-7
Anastasia Stavrova
{"title":"Isotropic reductive groups over discrete Hodge algebras","authors":"Anastasia Stavrova","doi":"10.1007/s40062-018-0221-7","DOIUrl":"https://doi.org/10.1007/s40062-018-0221-7","url":null,"abstract":"<p>Let <i>G</i> be a reductive group over a commutative ring <i>R</i>. We say that <i>G</i> has isotropic rank <span>(ge n)</span>, if every normal semisimple reductive <i>R</i>-subgroup of <i>G</i> contains <span>(({{mathrm{{{mathbf {G}}}_m}}}_{,R})^n)</span>. We prove that if <i>G</i> has isotropic rank <span>(ge 1)</span> and <i>R</i> is a regular domain containing an infinite field <i>k</i>, then for any discrete Hodge algebra <span>(A=R[x_1,ldots ,x_n]/I)</span> over <i>R</i>, the map <span>(H^1_{mathrm {Nis}}(A,G)rightarrow H^1_{mathrm {Nis}}(R,G))</span> induced by evaluation at <span>(x_1=cdots =x_n=0)</span>, is a bijection. If <i>k</i> has characteristic 0, then, moreover, the map <span>(H^1_{acute{mathrm{e}}mathrm {t}}(A,G)rightarrow H^1_{acute{mathrm{e}}mathrm {t}}(R,G))</span> has trivial kernel. We also prove that if <i>k</i> is perfect, <i>G</i> is defined over <i>k</i>, the isotropic rank of <i>G</i> is <span>(ge 2)</span>, and <i>A</i> is square-free, then <span>(K_1^G(A)=K_1^G(R))</span>, where <span>(K_1^G(R)=G(R)/E(R))</span> is the corresponding non-stable <span>(K_1)</span>-functor, also called the Whitehead group of <i>G</i>. The corresponding statements for <span>(G={{mathrm{GL}}}_n)</span> were previously proved by Ton Vorst.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"509 - 524"},"PeriodicalIF":0.5,"publicationDate":"2018-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0221-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4430486","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: 5
Koszuality of the (mathcal V^{(d)}) dioperad (mathcal V^{(d)}) dioperad的Koszuality
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-10-30 DOI: 10.1007/s40062-018-0220-8
Kate Poirier, Thomas Tradler
{"title":"Koszuality of the (mathcal V^{(d)}) dioperad","authors":"Kate Poirier,&nbsp;Thomas Tradler","doi":"10.1007/s40062-018-0220-8","DOIUrl":"https://doi.org/10.1007/s40062-018-0220-8","url":null,"abstract":"<p>Define a <span>(mathcal V^{(d)})</span>-algebra as an associative algebra with a symmetric and invariant co-inner product of degree <i>d</i>. Here, we consider <span>(mathcal V^{(d)})</span> as a dioperad which includes operations with zero inputs. We show that the quadratic dual of <span>(mathcal V^{(d)})</span> is <span>((mathcal V^{(d)})^!=mathcal V^{(-d)})</span> and prove that <span>(mathcal V^{(d)})</span> is Koszul. We also show that the corresponding properad is not Koszul contractible.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"477 - 507"},"PeriodicalIF":0.5,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0220-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5164922","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: 1
Delooping derived mapping spaces of bimodules over an operad 在操作符上开发双模的派生映射空间
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-10-17 DOI: 10.1007/s40062-018-0217-3
Julien Ducoulombier
{"title":"Delooping derived mapping spaces of bimodules over an operad","authors":"Julien Ducoulombier","doi":"10.1007/s40062-018-0217-3","DOIUrl":"https://doi.org/10.1007/s40062-018-0217-3","url":null,"abstract":"<p>To any topological operad <i>O</i>, we introduce a cofibrant replacement in the category of bimodules over itself such that for every map <span>(eta :Orightarrow O')</span> of operads, the corresponding model <span>({textit{Bimod}}_{O}^{h}(O,;,O'))</span> of derived mapping space of bimodules is an algebra over the one dimensional little cubes operad <span>(mathcal {C}_{1})</span>. We also build an explicit weak equivalence of <span>(mathcal {C}_{1})</span>-algebras from the loop space <span>(Omega {textit{Operad}}^{h}(O,;,O'))</span> to <span>({textit{Bimod}}_{O}^{h}(O,;,O'))</span>.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"411 - 453"},"PeriodicalIF":0.5,"publicationDate":"2018-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0217-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4700592","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
Homotopical algebraic context over differential operators 微分算子上的同局部代数环境
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-09-18 DOI: 10.1007/s40062-018-0213-7
Gennaro Di Brino, Damjan Pištalo, Norbert Poncin
{"title":"Homotopical algebraic context over differential operators","authors":"Gennaro Di Brino,&nbsp;Damjan Pištalo,&nbsp;Norbert Poncin","doi":"10.1007/s40062-018-0213-7","DOIUrl":"https://doi.org/10.1007/s40062-018-0213-7","url":null,"abstract":"<p>Building on our previous work, we show that the category of non-negatively graded chain complexes of <span>(mathcal {D}_X)</span>-modules – where <i>X</i> is a smooth affine algebraic variety over an algebraically closed field of characteristic zero – fits into a homotopical algebraic context in the sense of To?n and Vezzosi.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 1","pages":"293 - 347"},"PeriodicalIF":0.5,"publicationDate":"2018-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0213-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4737785","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: 11
Syntactic aspects of hypergraph polytopes 超图多面体的句法方面
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-08-13 DOI: 10.1007/s40062-018-0211-9
Pierre-Louis Curien, Jovana Obradović, Jelena Ivanović
{"title":"Syntactic aspects of hypergraph polytopes","authors":"Pierre-Louis Curien,&nbsp;Jovana Obradović,&nbsp;Jelena Ivanović","doi":"10.1007/s40062-018-0211-9","DOIUrl":"https://doi.org/10.1007/s40062-018-0211-9","url":null,"abstract":"<p>This paper introduces an inductive tree notation for all the faces of polytopes arising from a simplex by truncations, which allows viewing face inclusion as the process of contracting tree edges. These polytopes, known as hypergraph polytopes or nestohedra, fit in the interval from simplices to permutohedra (in any finite dimension). This interval was further stretched by Petri? to allow truncations of faces that are themselves obtained by truncations. Our notation applies to all these polytopes. As an illustration, we detail the case of Petri?’s permutohedron-based associahedra. As an application, we present a criterion for determining whether edges of polytopes associated with the coherences of categorified operads correspond to sequential, or to parallel associativity.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 1","pages":"235 - 279"},"PeriodicalIF":0.5,"publicationDate":"2018-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0211-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4522878","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: 13
Waldhausen Additivity: classical and quasicategorical 瓦尔德豪森可加性:经典和准范畴
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-07-12 DOI: 10.1007/s40062-018-0206-6
Thomas M. Fiore, Malte Pieper
{"title":"Waldhausen Additivity: classical and quasicategorical","authors":"Thomas M. Fiore,&nbsp;Malte Pieper","doi":"10.1007/s40062-018-0206-6","DOIUrl":"https://doi.org/10.1007/s40062-018-0206-6","url":null,"abstract":"<p>We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of <span>(wS_bullet )</span>, which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence <span>(wS_bullet {mathcal {E}}({mathcal {A}},{mathcal {C}},{mathcal {B}}) rightarrow wS_bullet {mathcal {A}}times wS_bullet {mathcal {B}})</span>. A consequence is Additivity for the Waldhausen <i>K</i>-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra <span>({mathbf {K}}({mathcal {A}}) vee {mathbf {K}}({mathcal {B}}) rightarrow {mathbf {K}}({mathcal {E}}({mathcal {A}},{mathcal {C}},{mathcal {B}})))</span>. This paper is dedicated to transferring these proofs to the quasicategorical setting and developing Waldhausen quasicategories and their sequences. We also give sufficient conditions for a split exact sequence to be equivalent to a standard one. These conditions are always satisfied by stable quasicategories, so Waldhausen <i>K</i>-theory sends any split exact sequence of pointed stable quasicategories to a split cofiber sequence. Presentability is not needed. In an effort to make the article self-contained, we recall all the necessary results from the theory of quasicategories, and prove a few quasicategorical results that are not in the literature.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 1","pages":"109 - 197"},"PeriodicalIF":0.5,"publicationDate":"2018-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0206-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4493150","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
On a Quillen adjunction between the categories of differential graded and simplicial coalgebras 微分阶代数与简单代数范畴间的一个Quillen附接
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-06-30 DOI: 10.1007/s40062-018-0210-x
W. Hermann B. Sore
{"title":"On a Quillen adjunction between the categories of differential graded and simplicial coalgebras","authors":"W. Hermann B. Sore","doi":"10.1007/s40062-018-0210-x","DOIUrl":"https://doi.org/10.1007/s40062-018-0210-x","url":null,"abstract":"<p>We prove that the normalization functor of the Dold-Kan correspondence does <i>not</i> induce a Quillen equivalence between Goerss’ model category of simplicial coalgebras and Getzler–Goerss’ model category of differential graded coalgebras.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 1","pages":"91 - 107"},"PeriodicalIF":0.5,"publicationDate":"2018-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0210-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5154421","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: 1
Model category of diffeological spaces 微分空间的模型范畴
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-06-20 DOI: 10.1007/s40062-018-0209-3
Hiroshi Kihara
{"title":"Model category of diffeological spaces","authors":"Hiroshi Kihara","doi":"10.1007/s40062-018-0209-3","DOIUrl":"https://doi.org/10.1007/s40062-018-0209-3","url":null,"abstract":"<p>The existence of a model structure on the category <span>({mathcal {D}})</span> of diffeological spaces is crucial to developing smooth homotopy theory. We construct a compactly generated model structure on the category <span>({mathcal {D}})</span> whose weak equivalences are just smooth maps inducing isomorphisms on smooth homotopy groups. The essential part of our construction of the model structure on <span>({mathcal {D}})</span> is to introduce diffeologies on the sets <span>(varDelta ^{p})</span><span>((p ge 0))</span> such that <span>(varDelta ^{p})</span> contains the <span>(kmathrm{th})</span> horn <span>(varLambda ^{p}_{k})</span> as a smooth deformation retract.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 1","pages":"51 - 90"},"PeriodicalIF":0.5,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0209-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4793919","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: 17
Frobenius pairs in abelian categories 阿贝尔范畴中的Frobenius对
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-05-17 DOI: 10.1007/s40062-018-0208-4
Víctor Becerril, Octavio Mendoza, Marco A. Pérez, Valente Santiago
{"title":"Frobenius pairs in abelian categories","authors":"Víctor Becerril,&nbsp;Octavio Mendoza,&nbsp;Marco A. Pérez,&nbsp;Valente Santiago","doi":"10.1007/s40062-018-0208-4","DOIUrl":"https://doi.org/10.1007/s40062-018-0208-4","url":null,"abstract":"<p>We revisit Auslander–Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures. From the notion of relative generators, we introduce the concept of left Frobenius pairs <span>(({mathcal {X}},omega ))</span> in an abelian category <span>({mathcal {C}})</span>. We show how to construct from <span>(({mathcal {X}},omega ))</span> a projective exact model structure on <span>({mathcal {X}}^wedge )</span>, the subcategory of objects in <span>({mathcal {C}})</span> with finite <span>({mathcal {X}})</span>-resolution dimension, via cotorsion pairs relative to a thick subcategory of <span>({mathcal {C}})</span>. We also establish correspondences between these model structures, relative cotorsion pairs, Frobenius pairs, and Auslander–Buchweitz contexts. Some applications of this theory are given in the context of Gorenstein homological algebra, and connections with perfect cotorsion pairs, covering subcategories and cotilting modules are also presented and described.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 1","pages":"1 - 50"},"PeriodicalIF":0.5,"publicationDate":"2018-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0208-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4693545","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: 23
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