{"title":"更正：关于 E $E$ 理论的拓扑学","authors":"José R. Carrión, Christopher Schafhauser","doi":"10.1112/jlms.70029","DOIUrl":null,"url":null,"abstract":"<p>The second sentence of [<span>1</span>, Corollary 4.4] does not follow from the given reference, and we do not know if it is true as stated. What is true is that if <span></span><math>\n <semantics>\n <mrow>\n <mover>\n <mi>x</mi>\n <mo>¯</mo>\n </mover>\n <mo>∈</mo>\n <msub>\n <mrow>\n <mo>[</mo>\n <mrow>\n <mo>[</mo>\n <mi>A</mi>\n <mo>,</mo>\n <mi>B</mi>\n <mo>]</mo>\n </mrow>\n <mo>]</mo>\n </mrow>\n <mi>Hd</mi>\n </msub>\n </mrow>\n <annotation>$\\bar{x} \\in [[A, B]]_{\\mathrm{Hd}}$</annotation>\n </semantics></math> is an isomorphism, then there is an isomorphism <span></span><math>\n <semantics>\n <mrow>\n <mi>x</mi>\n <mo>∈</mo>\n <mo>[</mo>\n <mo>[</mo>\n <mi>A</mi>\n <mo>,</mo>\n <mi>B</mi>\n <mo>]</mo>\n <mo>]</mo>\n </mrow>\n <annotation>$x \\in [[A, B]]$</annotation>\n </semantics></math> such that <span></span><math>\n <semantics>\n <mrow>\n <mi>Hd</mi>\n <mrow>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <mover>\n <mi>x</mi>\n <mo>¯</mo>\n </mover>\n </mrow>\n <annotation>$\\mathrm{Hd}(x) = \\bar{x}$</annotation>\n </semantics></math>. Indeed, [<span>2</span>, Theorem 1.14] implies every isomorphism in the shape category <span></span><math>\n <semantics>\n <mi>sh</mi>\n <annotation>$\\mathsf {sh}$</annotation>\n </semantics></math> is induced by an isomorphism in the strong shape category <span></span><math>\n <semantics>\n <mi>s</mi>\n <annotation>$\\mathsf {s}$</annotation>\n </semantics></math>-<span></span><math>\n <semantics>\n <mi>sh</mi>\n <annotation>$\\mathsf {sh}$</annotation>\n </semantics></math>, and then the result follows from using [<span>1</span>, Theorem 4.3; <span>2</span>, Theorem 3.7] to identify these categories with the Hausdorffized asymptotic morphism category <span></span><math>\n <semantics>\n <msub>\n <mi>AM</mi>\n <mi>Hd</mi>\n </msub>\n <annotation>$\\mathsf {AM}_{\\mathrm{Hd}}$</annotation>\n </semantics></math> and the asymptotic morphism category <span></span><math>\n <semantics>\n <mi>AM</mi>\n <annotation>$\\mathsf {AM}$</annotation>\n </semantics></math>.</p><p>This error has no effect on the rest of the results in the paper.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70029","citationCount":"0","resultStr":"{\"title\":\"Corrigendum: A topology on \\n \\n E\\n $E$\\n -theory\",\"authors\":\"José R. Carrión, Christopher Schafhauser\",\"doi\":\"10.1112/jlms.70029\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The second sentence of [<span>1</span>, Corollary 4.4] does not follow from the given reference, and we do not know if it is true as stated. What is true is that if <span></span><math>\\n <semantics>\\n <mrow>\\n <mover>\\n <mi>x</mi>\\n <mo>¯</mo>\\n </mover>\\n <mo>∈</mo>\\n <msub>\\n <mrow>\\n <mo>[</mo>\\n <mrow>\\n <mo>[</mo>\\n <mi>A</mi>\\n <mo>,</mo>\\n <mi>B</mi>\\n <mo>]</mo>\\n </mrow>\\n <mo>]</mo>\\n </mrow>\\n <mi>Hd</mi>\\n </msub>\\n </mrow>\\n <annotation>$\\\\bar{x} \\\\in [[A, B]]_{\\\\mathrm{Hd}}$</annotation>\\n </semantics></math> is an isomorphism, then there is an isomorphism <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>x</mi>\\n <mo>∈</mo>\\n <mo>[</mo>\\n <mo>[</mo>\\n <mi>A</mi>\\n <mo>,</mo>\\n <mi>B</mi>\\n <mo>]</mo>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$x \\\\in [[A, B]]$</annotation>\\n </semantics></math> such that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Hd</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>=</mo>\\n <mover>\\n <mi>x</mi>\\n <mo>¯</mo>\\n </mover>\\n </mrow>\\n <annotation>$\\\\mathrm{Hd}(x) = \\\\bar{x}$</annotation>\\n </semantics></math>. Indeed, [<span>2</span>, Theorem 1.14] implies every isomorphism in the shape category <span></span><math>\\n <semantics>\\n <mi>sh</mi>\\n <annotation>$\\\\mathsf {sh}$</annotation>\\n </semantics></math> is induced by an isomorphism in the strong shape category <span></span><math>\\n <semantics>\\n <mi>s</mi>\\n <annotation>$\\\\mathsf {s}$</annotation>\\n </semantics></math>-<span></span><math>\\n <semantics>\\n <mi>sh</mi>\\n <annotation>$\\\\mathsf {sh}$</annotation>\\n </semantics></math>, and then the result follows from using [<span>1</span>, Theorem 4.3; <span>2</span>, Theorem 3.7] to identify these categories with the Hausdorffized asymptotic morphism category <span></span><math>\\n <semantics>\\n <msub>\\n <mi>AM</mi>\\n <mi>Hd</mi>\\n </msub>\\n <annotation>$\\\\mathsf {AM}_{\\\\mathrm{Hd}}$</annotation>\\n </semantics></math> and the asymptotic morphism category <span></span><math>\\n <semantics>\\n <mi>AM</mi>\\n <annotation>$\\\\mathsf {AM}$</annotation>\\n </semantics></math>.</p><p>This error has no effect on the rest of the results in the paper.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-11-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70029\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70029\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70029","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
[1，推论 4.4] 的第二句话并不是从给出的参考文献中得出的，我们也不知道它是否如所说的那样是真的。真实的情况是，如果 x ∈ [ [ A , B ] ] Hd $\bar{x}\in [[A, B]]_{mathrm{Hd}}$ 是一个同构，那么就有一个同构 x ∈ [ [ A , B ] ]。 ] $x \in [[A, B]]$ 这样 Hd ( x ) = x ¯ $\mathrm{Hd}(x) = \bar{x}$ 。事实上，[2, Theorem 1.14]意味着形状范畴 sh $\mathsf {sh}$ 中的每一个同构都是由强形状范畴 s $\mathsf {s}$ - sh $\mathsf {sh}$ 中的一个同构诱导的，然后使用[1, Theorem 4.3; 2, Theorem 3.7] 将这些范畴与 Hausdorffized渐近形态范畴 AM Hd $\mathsf {AM}_{\mathrm{Hd}}$ 和渐近形态范畴 AM $\mathsf {AM}$ 标识开来，就得出了结果。这个错误对本文的其他结果没有影响。
The second sentence of [1, Corollary 4.4] does not follow from the given reference, and we do not know if it is true as stated. What is true is that if is an isomorphism, then there is an isomorphism such that . Indeed, [2, Theorem 1.14] implies every isomorphism in the shape category is induced by an isomorphism in the strong shape category -, and then the result follows from using [1, Theorem 4.3; 2, Theorem 3.7] to identify these categories with the Hausdorffized asymptotic morphism category and the asymptotic morphism category .
This error has no effect on the rest of the results in the paper.
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The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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