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type="application/xml" title="MHuiG Blog Site Map" href="https://blog.mhuig.top/sitemap.xml"><link rel="author" href="https://mhuig.top"><meta name="author" content="MHuiG"><meta name="creator" content="MHuiG"><link rel="archives" href="https://blog.mhuig.top/archives/"><link rel="preload" href="/css/style.css" as="style"><link rel="preload" href="https://static.mhuig.top/npm/volantis-static@0.0.1660614606622/media/fonts/VarelaRound/VarelaRound-Regular.ttf" as="font" type="font/ttf" crossorigin="anonymous"><link rel="preload" href="https://static.mhuig.top/npm/volantis-static@0.0.1660614606622/media/fonts/VarelaRound/VarelaRound-Regular.ttf" as="font" type="font/ttf" crossorigin="anonymous"><link rel="alternate" href="/atom.xml" title="Magicland" type="application/atom+xml"><link rel="alternate" href="/rss2.xml" title="Magicland" type="application/rss+xml"><title>History of mathematics: Waldhausen瓦尔德豪森的数学人生：从三维流形解剖到美丽新代数的开创 - Magicland</title><meta name="keywords" content="历史,Person, History,人物,瓦尔德豪森,三维流形,代数K理论,Waldhausen范畴,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc="" name="description" content="本文记述了德国数学家瓦尔德豪森的生涯与成就。他早期在三维流形理论中建立了拓扑刚性定理并提出了著名的virtual Haken猜想；后期开创了用于研究拓扑空间的代数K理论框架，包括Waldhausen范畴和S_*构造，其思想深刻影响了稳定同伦论等前沿领域 - MHuiG - Magicland"><meta property="og:type" content="website"><meta property="og:title" content="Magicland"><meta property="og:url" content="https://blog.mhuig.top/notes/person/6"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="本文记述了德国数学家瓦尔德豪森的生涯与成就。他早期在三维流形理论中建立了拓扑刚性定理并提出了著名的virtual Haken猜想；后期开创了用于研究拓扑空间的代数K理论框架，包括Waldhausen范畴和S_*构造，其思想深刻影响了稳定同伦论等前沿领域"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2026-04-01T03:07:00.000Z"><meta property="article:modified_time" content="2026-05-03T03:10:00.000Z"><meta property="article:author" content="MHuiG"><meta 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shadow floatable blur article-type-docs" id="docs" itemprop="blogPost"><link itemprop="mainEntityOfPage" href="https://blog.mhuig.top/notes/person/6"><span hidden="" itemprop="publisher" itemscope="" itemtype="http://schema.org/Organization"><meta itemprop="name" content="Magicland"></span><span hidden="" itemprop="post" itemscope="" itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="Waldhausen瓦尔德豪森的数学人生：从三维流形解剖到美丽新代数的开创"><meta itemprop="description" content="本文记述了德国数学家瓦尔德豪森的生涯与成就。他早期在三维流形理论中建立了拓扑刚性定理并提出了著名的virtual Haken猜想；后期开创了用于研究拓扑空间的代数K理论框架，包括Waldhausen范畴和S_*构造，其思想深刻影响了稳定同伦论等前沿领域"></span><span hidden=""><meta itemprop="image" content="/lib/favicon/android-chrome-192x192.png"></span><div class="article-meta" id="top"><span hidden="" itemprop="name headline"></span></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p> <span class="p logo center large">Waldhausen</span></p><br><h1 hidden="">Waldhausen</h1><div class="story post-story"><h2 id="生平"><a href="#生平" class="headerlink" title="生平"></a>生平</h2><p>1971 年 4 月 1 日，Waldhausen 正式加入 Bielefeld 大学担任全职教授，并在此度过了他漫长而辉煌的执教生涯，直到 2004 年退休。</p><p>在此期间，他两度出任数学系主任（1973/74 和 1984/85），不仅一手塑造了该校拓扑学研究团队的国际声望，还培养了包括斯威德（Stefan Schwede）和伦迪格斯（Oliver Rön digs）在内的一批优秀学者。</p><p>他的博士生许特曼（Thomas Huettemann）回忆说：<br>“Waldhausen 曾给过学生两条受用终身的务实建议。第一，在写数学论文时，永远不要使用‘容易’或者‘显然’这样的词汇。第二，阅读一本数学书时，应该‘倒着读’，也就是直接从书里那个最吸引人的有趣结果开始往前探索。”</p><p>他的学生 Stefan Schwede 回忆： <strong>Waldhausen 讲授的高级课程极其硬核</strong> ，随着学期推进，退课的学生越来越多，到最后往往只剩下两名学生。但令人敬佩的是，只要剩下的学生还愿意学， <strong>Waldhausen 就会彻底无视学期之间的寒暑假和法定休息日，一直讲下去</strong> 。直到最后唯一的学生也决定退出时，这门课才会宣告结束。</p><p>2004 年，为了表彰他在纯数学领域的卓越成就，他与哈德尔（Günter Harder，1938-2025）共同获得了德国数学界最负盛名的奖项之一 —— <strong>von Staudt Prize</strong> 。此外，他还获得了 Osnabrück 大学授予的荣誉博士学位。</p><p>von Staudt Prize 获奖者</p><table><thead><tr><th>获奖时间</th><th>获奖人</th><th>获奖时间</th><th>获奖人</th></tr></thead><tbody><tr><td>1991</td><td>H. Grauert</td><td>1994</td><td>S. Hildebrandt</td></tr><tr><td>1997</td><td>M. Kneser</td><td>2001</td><td>D. Zagier</td></tr><tr><td>2004</td><td>G. Harder<br>F. Waldhausen</td><td>2008</td><td>G. Faltings</td></tr><tr><td>2013</td><td>M. Rapoport</td><td>2022</td><td>B. Wilking</td></tr><tr><td>2025</td><td>W. Lück</td><td></td><td></td></tr></tbody></table><p>在生活中，他有着独特的个人爱好，例如热爱舒伯特的艺术歌曲（Schubert lieder）以及偏爱加大蒜的美食。</p><p>2024 年 11 月 21 日，Friedhelm Waldhausen 逝世，享年 85 岁，留下了极其丰富的数学遗产。</p></div><div class="story post-story"><h2 id="贡献"><a href="#贡献" class="headerlink" title="贡献"></a>贡献</h2><h3 id="一、3-维流形理论：空间的“解剖学”与“代数刚性”"><a href="#一、3-维流形理论：空间的“解剖学”与“代数刚性”" class="headerlink" title="一、3 维流形理论：空间的“解剖学”与“代数刚性”"></a>一、3 维流形理论：空间的 “解剖学” 与 “代数刚性”</h3><p>在他的学术早期，Waldhausen 的核心思想可以概括为：如何通过 “内部切片” 来理解整体空间，以及空间的 “代数骨架” 如何绝对地决定它的 “几何肉体”。</p><h4 id="1、Haken-流形与不可压缩曲面（incompressible-surfaces）"><a href="#1、Haken-流形与不可压缩曲面（incompressible-surfaces）" class="headerlink" title="1、Haken 流形与不可压缩曲面（incompressible surfaces）"></a>1、Haken 流形与不可压缩曲面（incompressible surfaces）</h4><p>在哈肯（Wolfgang Haken，1928-2022）等的基础上，Waldhausen 重点研究了一类包含内部结构的 3 维流形，即后来的 Haken 流形。</p><p>理解它的核心在于 “不可压缩（incompressible）” 性。</p><p>在拓扑学中，我们用<strong>基本群</strong>中的闭合环路（loop）来探测空间的孔洞。假设 3 维流形<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="2.362ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1044 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="M"><g data-mml-node="mi" data-latex="M"><path data-c="1D440" d="M594 16C594 5 600 0 613 0L736 3C759 4 840 0 859 0 874 0 882 8 882 24 882 34 872 39 851 39 810 39 790 43 790 52 790 55 792 62 795 75L927 602C932 623 942 636 955 641 961 643 979 644 1008 644 1033 644 1044 645 1044 668 1044 678 1034 683 1013 683L882 683C853 683 853 681 840 662L484 108 408 657C404 682 402 683 373 683L237 683C215 683 204 682 204 660 204 649 215 644 236 644 256 644 297 647 297 631 297 629 296 623 293 613L167 110C158 72 135 49 100 42 75 39 42 43 42 16 42 5 48 0 60 0 78 0 141 3 159 3 178 3 242 0 261 0 276 0 283 8 283 24 283 33 276 38 261 39 219 39 198 52 198 78 198 82 199 89 202 100L332 621 415 26C418 9 424 0 434 0 442 0 450 7 459 20L848 627 712 82C706 60 696 47 681 42 675 40 656 39 625 39 604 39 594 37 594 16Z"></path></g></g></g></svg></mjx-container> 内部飘浮着一个 2 维闭曲面<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.459ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 645 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="S"><g data-mml-node="mi" data-latex="S"><path data-c="1D446" d="M133 157C133 181 136 201 141 217 141 227 136 232 125 232 120 232 116 230 114 228 109 223 52 8 52-8 52-17 57-22 67-22 72-22 79-17 88-6L133 47C168 1 224-22 300-22 366-22 424 5 476 58 528 111 554 170 554 236 554 283 538 323 505 355 490 368 470 379 445 388 422 393 400 399 378 405L312 423C279 432 254 469 254 509 254 552 271 589 306 621 341 653 380 669 423 669 515 669 561 619 561 520 561 502 557 482 557 465L557 462C560 455 565 451 573 451 582 451 588 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423C279 432 254 469 254 509 254 552 271 589 306 621 341 653 380 669 423 669 515 669 561 619 561 520 561 502 557 482 557 465L557 462C560 455 565 451 573 451 582 451 588 459 592 474L645 691C645 700 640 705 630 705 625 705 618 700 609 689L566 637C538 682 491 705 424 705 361 705 304 681 254 634 202 586 176 531 176 468 176 396 223 339 282 323L387 296C441 281 475 262 475 195 475 149 457 108 422 72 387 36 347 17 302 17 204 17 133 60 133 157Z"></path></g></g></g></svg></mjx-container> 是不可压缩的，意味着：只要在曲面<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.459ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 645 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="S"><g data-mml-node="mi" data-latex="S"><path data-c="1D446" d="M133 157C133 181 136 201 141 217 141 227 136 232 125 232 120 232 116 230 114 228 109 223 52 8 52-8 52-17 57-22 67-22 72-22 79-17 88-6L133 47C168 1 224-22 300-22 366-22 424 5 476 58 528 111 554 170 554 236 554 283 538 323 505 355 490 368 470 379 445 388 422 393 400 399 378 405L312 423C279 432 254 469 254 509 254 552 271 589 306 621 341 653 380 669 423 669 515 669 561 619 561 520 561 502 557 482 557 465L557 462C560 455 565 451 573 451 582 451 588 459 592 474L645 691C645 700 640 705 630 705 625 705 618 700 609 689L566 637C538 682 491 705 424 705 361 705 304 681 254 634 202 586 176 531 176 468 176 396 223 339 282 323L387 296C441 281 475 262 475 195 475 149 457 108 422 72 387 36 347 17 302 17 204 17 133 60 133 157Z"></path></g></g></g></svg></mjx-container> 上画的绳圈能在外部的 3 维空间<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="2.362ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1044 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="M"><g data-mml-node="mi" data-latex="M"><path data-c="1D440" d="M594 16C594 5 600 0 613 0L736 3C759 4 840 0 859 0 874 0 882 8 882 24 882 34 872 39 851 39 810 39 790 43 790 52 790 55 792 62 795 75L927 602C932 623 942 636 955 641 961 643 979 644 1008 644 1033 644 1044 645 1044 668 1044 678 1034 683 1013 683L882 683C853 683 853 681 840 662L484 108 408 657C404 682 402 683 373 683L237 683C215 683 204 682 204 660 204 649 215 644 236 644 256 644 297 647 297 631 297 629 296 623 293 613L167 110C158 72 135 49 100 42 75 39 42 43 42 16 42 5 48 0 60 0 78 0 141 3 159 3 178 3 242 0 261 0 276 0 283 8 283 24 283 33 276 38 261 39 219 39 198 52 198 78 198 82 199 89 202 100L332 621 415 26C418 9 424 0 434 0 442 0 450 7 459 20L848 627 712 82C706 60 696 47 681 42 675 40 656 39 625 39 604 39 594 37 594 16Z"></path></g></g></g></svg></mjx-container> 中收缩成一个点，那么它必须在曲面<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.459ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 645 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="S"><g data-mml-node="mi" data-latex="S"><path data-c="1D446" d="M133 157C133 181 136 201 141 217 141 227 136 232 125 232 120 232 116 230 114 228 109 223 52 8 52-8 52-17 57-22 67-22 72-22 79-17 88-6L133 47C168 1 224-22 300-22 366-22 424 5 476 58 528 111 554 170 554 236 554 283 538 323 505 355 490 368 470 379 445 388 422 393 400 399 378 405L312 423C279 432 254 469 254 509 254 552 271 589 306 621 341 653 380 669 423 669 515 669 561 619 561 520 561 502 557 482 557 465L557 462C560 455 565 451 573 451 582 451 588 459 592 474L645 691C645 700 640 705 630 705 625 705 618 700 609 689L566 637C538 682 491 705 424 705 361 705 304 681 254 634 202 586 176 531 176 468 176 396 223 339 282 323L387 296C441 281 475 262 475 195 475 149 457 108 422 72 387 36 347 17 302 17 204 17 133 60 133 157Z"></path></g></g></g></svg></mjx-container> 自身上也能收缩成一点。</p><p>用代数语言来说，就是曲面的基本群到流形基本群的诱导同态必须是单射，即<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="5.497ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2429.6 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\pi_1(S) \to \pi_1(M)"><g data-mml-node="msub" data-latex="\pi_1"><g data-mml-node="mi" data-latex="\pi"><path data-c="1D70B" d="M524 431 194 431C153 431 117 415 88 384 74 369 27 305 27 292 31 285 31 279 43 279 50 279 56 283 62 292 94 341 135 366 184 366L235 366C212 279 170 172 110 45 105 33 102 24 102 19 102-1 113-11 134-11 153-11 167 0 176 21 194 78 207 122 214 152L269 366 372 366C345 247 331 164 331 117 331 68 342-11 379-11 400-11 423 9 423 30 423 35 421 43 417 53 398 100 389 153 389 214 389 261 395 312 406 366L515 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href="#2、拓扑刚性定理（Topological-Rigidity-Theorem-1968）" class="headerlink" title="2、拓扑刚性定理（Topological Rigidity Theorem, 1968）"></a>2、拓扑刚性定理（Topological Rigidity Theorem, 1968）</h4><p>在拓扑学中，同伦等价（homotopy equivalence）是一种较弱的 “代数形变” 关系（允许空间被无限拉伸、压扁，只要孔洞等代数特征不变即可），而同胚（homeomorphism）则是最强的 “几何全等” 关系（要求两个空间点对点完美匹配，绝不能撕裂或粘合）。</p><p>通常，代数相等不代表几何相等（高维空间有大量反例）。</p><p>但 Waldhausen 证明了极其震撼的结论：对于闭的、可定向的 Haken 流形，“代数完全决定了几何”：</p><p>对于闭的、可定向的 Haken 流形，任意同伦等价都同伦于一个同胚。因而，在这类流形中，拓扑结构在很强意义下由基本群（代数结构）所决定。</p><h4 id="3、Heegaard-分解与-virtual-Haken-conjecture"><a href="#3、Heegaard-分解与-virtual-Haken-conjecture" class="headerlink" title="3、Heegaard 分解与 virtual Haken conjecture"></a>3、Heegaard 分解与 virtual Haken conjecture</h4><p>Heegaard 分解（Heegaard splitting）是将任何 3 维流形切成两个柄体（handlebodies），然后沿着它们的表面重新缝合。</p><p>Waldhausen 证明了对于我们最熟悉的 3 - 维球面<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.027ex" xmlns="http://www.w3.org/2000/svg" width="2.243ex" height="1.914ex" role="img" 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348 88 317 14 241 14 176 14 128 36 99 80 134 79 160 105 160 139 160 173 135 198 101 198 62 198 42 178 42 137 42 88 64 49 108 18 147-9 193-22 244-22 301-22 350-3 393 34 436 71 457 117 457 173 457 267 383 332 303 353Z"></path></g></g></g></g></svg></mjx-container> ，所有相同亏格（洞的数量）的 Heegaard 分解，在同痕（isotopy，即连续形变）意义下都是唯一的。这排除了基础空间中存在 “病态” 分解的可能。</p><p>由于并非所有 3 维流形内部都能找到不可压缩曲面（即并非都是 Haken 流形），Waldhausen 提出了著名的 <strong>virtual Haken conjecture</strong>：任何紧致、不可约且基本群无限的 3 维流形，都应该存在一个有限张覆盖空间（covering space），使得这个覆盖空间成为 Haken 流形。</p><p>也就是说，虽然原来的流形未必直接含有不可压缩曲面，但在适当的有限覆盖中，这种 “隐藏的内部结构” 会显现出来。</p><p>这个猜想悬置了 40 多年，最终在 2012 年由阿戈尔（Ian Agol, 1970-）利用前沿的几何群论彻底证明。</p><h3 id="二、代数-K-理论与美丽新代数（brave-new-algebra）"><a href="#二、代数-K-理论与美丽新代数（brave-new-algebra）" class="headerlink" title="二、代数 K-理论与美丽新代数（brave new algebra）"></a>二、代数 K- 理论与美丽新代数（brave new algebra）</h3><p>到了 1970 年代中期，Waldhausen 意识到传统几何工具已经无法处理更高维流形的复杂形变。为此，他发明了一套全新的抽象代数机器。</p><h4 id="1、空间的代数-K-理论"><a href="#1、空间的代数-K-理论" class="headerlink" title="1、空间的代数 K-理论"></a>1、空间的代数 K- 理论</h4><p>代数 K- 理论（由 Grothendieck, Atiyah, Hirzebruch 等创立，Quillen 奠定高阶基础）研究环和概形等纯代数结构。</p><p>为了研究流形的伪同痕空间（pseudo-isotopy space）和微分同胚群（diffeomorphism groups），Waldhausen 创造性地将输入端从 “代数环” 替换成了 “拓扑空间<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.925ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 851 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="X"><g data-mml-node="mi" data-latex="X"><path data-c="1D44B" d="M834 683C817 683 758 680 741 680 721 680 651 683 631 683 616 683 608 675 608 659 608 650 613 645 624 644 648 641 660 633 660 618 660 609 654 598 642 585L486 417C457 485 429 552 400 620 401 622 402 624 405 626 414 636 431 642 455 644 471 646 479 654 479 667 479 678 473 683 460 683 436 683 359 680 335 680 314 680 245 683 224 683 209 683 202 675 202 659 202 649 211 644 229 644 274 644 288 644 300 615L416 342 208 119C205 116 201 112 196 109 155 66 108 42 53 39 36 38 27 29 27 14 30 5 31 0 44 0 61 0 120 3 137 3 158 3 227 0 248 0 263 0 270 8 270 24 270 33 265 38 254 39 230 41 218 50 218 65 218 75 227 89 244 107 306 173 369 239 430 306 464 225 499 145 532 63 531 61 530 59 527 56 518 47 502 42 478 39 463 37 455 29 455 16 455 5 461 0 474 0L599 3C620 3 690 0 708 0 723 0 731 8 731 23 731 33 724 38 709 39 678 39 658 40 651 44 644 48 636 60 628 80L501 383C569 455 620 509 653 546 677 571 693 588 702 595 739 626 780 642 825 644 846 645 851 650 851 669 848 678 847 683 834 683Z"></path></g></g></g></svg></mjx-container> ”，发明了<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg></mjx-container> 理论（<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg></mjx-container> -theory,<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.561ex" xmlns="http://www.w3.org/2000/svg" width="5.382ex" height="2.253ex" role="img" focusable="false" viewBox="0 -748 2379 996"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A(X)"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g><g data-mml-node="mo" data-latex="(" transform="translate(750,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 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transform="translate(646,-150) scale(0.707)" data-latex="*"><path data-c="2217" d="M404 372C397 372 390 370 385 366L267 279 282 425C285 445 269 462 250 462 231 462 215 445 218 425L233 279 115 366C110 370 103 372 96 372 78 372 63 357 63 339 63 325 70 315 83 309L217 250 83 191C70 185 63 175 63 161 63 143 78 128 96 128 103 128 110 130 115 134L233 221 218 75C215 55 231 39 250 39 269 39 285 55 282 75L267 221 385 134C390 130 397 128 404 128 415 128 424 133 431 142 446 162 435 183 417 191L283 250 417 309C430 315 437 325 437 339 437 357 422 372 404 372Z"></path></g></g></g></g></svg></mjx-container> 构造</h4><p>Quillen（1940-2011）原有的代数机器依赖于正合范畴（exact categories），这要求底层的数学对象必须能像数字或多项式那样进行严格的 “相加”（Abelian 环境）。</p><p>但是，拓扑空间是不能简单相加的（它们是非加性的）。Waldhausen 抛弃了传统的加法限制，发明了一套极简的底层逻辑框架 ——Waldhausen 范畴（Waldhausen categories）。</p><p>在这个世界里，他只保留了两个最具拓扑意味的动作：</p><p><strong>上纤维化（cofibrations）</strong> （允许将一个小空间 “干净地” 嵌入并添加到大空间中）和 <strong>弱等价（weak equivalences）</strong> （把形状极其相似、可连续形变的空间视作等同）。</p><p>基于此，他发明了令人惊叹的<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.277ex" xmlns="http://www.w3.org/2000/svg" width="2.375ex" height="1.872ex" role="img" focusable="false" viewBox="0 -705 1049.6 827.4"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="S_*"><g data-mml-node="msub" data-latex="S_*"><g data-mml-node="mi" data-latex="S"><path data-c="1D446" d="M133 157C133 181 136 201 141 217 141 227 136 232 125 232 120 232 116 230 114 228 109 223 52 8 52-8 52-17 57-22 67-22 72-22 79-17 88-6L133 47C168 1 224-22 300-22 366-22 424 5 476 58 528 111 554 170 554 236 554 283 538 323 505 355 490 368 470 379 445 388 422 393 400 399 378 405L312 423C279 432 254 469 254 509 254 552 271 589 306 621 341 653 380 669 423 669 515 669 561 619 561 520 561 502 557 482 557 465L557 462C560 455 565 451 573 451 582 451 588 459 592 474L645 691C645 700 640 705 630 705 625 705 618 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0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>及其商空间，像搭积木一样编织出一个多维的 <strong>单纯集合（simplicial set）</strong> 。这个集合的同伦群就是我们要找的<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="2.011ex" height="1.548ex" role="img" focusable="false" viewBox="0 -684 889 684"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="K"><g data-mml-node="mi" data-latex="K"><path data-c="1D43E" d="M804 680C783 680 715 683 694 683 679 683 671 675 671 659 671 651 678 646 692 643 706 640 714 635 714 628 714 617 703 602 680 585L302 296 379 602C384 623 393 636 406 641 412 643 429 644 458 644 484 644 496 644 496 668 496 678 490 683 478 683L351 680 223 683C207 684 199 676 199 660 199 649 210 644 232 644 252 644 293 647 293 631 293 629 292 623 289 613L156 82C151 60 141 47 126 42 119 40 101 39 70 39 48 39 38 37 38 16 38 5 44 0 57 0L183 3 246 2C257 2 299 0 310 0 326 0 334 8 334 24 334 34 324 39 303 39 262 39 242 43 242 52 242 57 245 72 251 95L290 249 433 359 546 98C553 81 557 70 557 65 557 48 541 39 510 39 492 39 484 35 484 15 484 5 490 0 502 0 524 0 600 3 622 3 641 3 701 0 720 0 734 0 741 8 741 24 741 33 733 38 718 39 677 39 667 50 653 81L512 404C511 409 509 413 508 416L708 570C757 608 799 631 833 639 857 643 889 642 889 668 889 678 884 683 874 683 860 683 818 680 804 680Z"></path></g></g></g></svg></mjx-container> - 群。</p><h4 id="3、美丽新代数（Brave-new-algebra）"><a href="#3、美丽新代数（Brave-new-algebra）" class="headerlink" title="3、美丽新代数（Brave new algebra）"></a>3、美丽新代数（Brave new algebra）</h4><p>这是 Waldhausen 晚年对纯数学的一个极具诗意且精准的概括（这个命名借用了 Huxley 的小说名）。在传统的初等代数中，一切都建立在整数环<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.507ex" height="1.55ex" role="img" focusable="false" viewBox="0 -685 666 685"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\mathbb{Z}"><g data-mml-node="TeXAtom" data-latex="\mathbb{Z}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="Z"><path data-c="2124" d="M569 685 117 685C103 685 93 684 88 681 83 670 79 661 78 653 73 604 68 554 62 506 62 491 69 483 83 483 94 483 101 489 104 501 119 594 217 642 332 642L405 642 37 42C31 31 28 25 28 22 28 4 43 0 66 0L572 0C601 0 607 5 609 31L636 226C636 241 629 248 615 248 610 248 605 246 602 243 596 231 593 222 592 215 571 117 480 43 362 43L228 43 596 643C602 654 605 660 605 663 605 682 593 685 569 685M156 642C144 635 131 625 116 612L119 642M455 642 546 642 179 43 87 43M517 43C538 56 558 73 575 92L568 43Z"></path></g></g></g></g></svg></mjx-container> 的基础上，等号<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:.301ex" xmlns="http://www.w3.org/2000/svg" width="1.76ex" height="0.529ex" role="img" focusable="false" viewBox="0 -367 778 234"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="="><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367 80 367C64 367 56 359 56 344 56 329 64 321 80 321L698 321C714 321 722 329 722 344 722 356 711 367 698 367M698 179 80 179C64 179 56 171 56 156 56 141 64 133 80 133L698 133C714 133 722 141 722 156 722 169 711 179 698 179Z"></path></g></g></g></svg></mjx-container> 是绝对严格的（比如结合律<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="4.907ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2169 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="(ab)c = a(bc)"><g data-mml-node="mo" data-latex="("><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mi" data-latex="a" transform="translate(389,0)"><path data-c="1D44E" d="M498 144C498 153 493 158 482 158 474 158 468 151 465 137 445 58 421 18 394 18 377 18 368 32 368 60 368 73 372 97 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216 442 159 409 109 343 63 281 40 217 40 150 40 63 91-11 175-11 218-11 260 12 300 58 311 20 345-11 392-11 461-11 482 70 498 144M341 374C350 353 355 339 355 330 355 326 354 321 353 314L304 122C301 111 294 99 285 87 248 41 212 18 177 18 138 18 118 48 118 107 118 131 124 167 136 215 157 300 187 357 224 388 244 405 263 413 282 413 309 413 329 400 341 374Z"></path></g><g data-mml-node="mo" data-latex="(" transform="translate(1584.8,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mi" data-latex="b" transform="translate(1973.8,0)"><path data-c="1D44F" d="M281 445C245 445 209 428 174 394L243 679C241 688 238 694 226 694 193 694 120 685 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transform="translate(2835.8,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> 必须绝对相等）。</p><p>但 Waldhausen 发现，如果要处理拓扑空间，我们需要彻底改写代数的基石：用具有弹性的球面谱（sphere spectrum,<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.027ex" xmlns="http://www.w3.org/2000/svg" width="1.256ex" height="1.62ex" role="img" focusable="false" viewBox="0 -704 555 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\mathbb{S}"><g data-mml-node="TeXAtom" data-latex="\mathbb{S}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="S"><path data-c="1D54A" d="M258 661C296 661 334 644 371 612 408 580 431 545 440 507 444 489 451 480 462 480 476 480 483 492 483 517L483 667C483 692 476 704 462 704 458 704 452 701 444 695 431 682 416 676 399 676 384 676 362 681 332 690 302 699 276 704 256 704 120 704 29 635 29 502 29 464 42 429 67 396 85 373 114 350 155 329 162 326 196 310 259 281 270 276 285 267 305 256 338 236 361 212 376 185 384 170 388 154 388 137 388 58 346 31 264 31 218 31 176 52 137 93 98 134 79 178 79 224 79 238 72 245 59 245 41 245 37 231 37 209L37 25C37 0 44-12 59-12 62-12 70-7 81 2 92 11 106 16 121 16 138 16 156 12 177 3 202-7 231-12 263-12 334-12 395 4 444 36 501 74 530 127 530 194 530 297 466 374 337 425 286 444 219 474 192 498 167 519 154 542 154 565 154 624 199 661 258 661M440 640 440 600C433 609 423 620 410 634 423 635 433 637 440 640M125 626C116 607 112 587 112 565 112 521 135 484 180 454 211 434 255 413 311 390 370 365 412 341 435 317 470 279 488 238 488 194 488 140 465 99 420 71 427 90 431 112 431 137 431 188 410 231 367 264 334 289 295 312 249 332 194 355 153 377 126 397 89 424 71 459 71 502 71 559 89 601 125 626M79 96C86 85 97 73 112 58 99 57 88 55 79 51Z"></path></g></g></g></g></svg></mjx-container> ）取代整数<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.507ex" height="1.55ex" role="img" focusable="false" viewBox="0 -685 666 685"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\mathbb{Z}"><g data-mml-node="TeXAtom" data-latex="\mathbb{Z}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="Z"><path data-c="2124" d="M569 685 117 685C103 685 93 684 88 681 83 670 79 661 78 653 73 604 68 554 62 506 62 491 69 483 83 483 94 483 101 489 104 501 119 594 217 642 332 642L405 642 37 42C31 31 28 25 28 22 28 4 43 0 66 0L572 0C601 0 607 5 609 31L636 226C636 241 629 248 615 248 610 248 605 246 602 243 596 231 593 222 592 215 571 117 480 43 362 43L228 43 596 643C602 654 605 660 605 663 605 682 593 685 569 685M156 642C144 635 131 625 116 612L119 642M455 642 546 642 179 43 87 43M517 43C538 56 558 73 575 92L568 43Z"></path></g></g></g></g></svg></mjx-container> 。</p><p>在 “美丽新代数” 的世界里，数字变成了拓扑 “空间”。乘法结合律和交换律不再产生绝对严格的相等，而是 “存在一条连续的路径将它们连接起来”（即在同伦意义下相等）。</p><p>不仅如此，路径与路径之间还有更高维的面相连，面与面之间有体相接，形成了一套极其庞大的高阶相干性（higher coherence）。</p><p>他用 “美丽新代数” 来形容这些高度结构化的环谱<br>(highly structured ring spectra, 如<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.357ex" xmlns="http://www.w3.org/2000/svg" width="3.484ex" height="1.977ex" role="img" focusable="false" viewBox="0 -716 1540.1 873.8"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A_\infty"><g data-mml-node="msub" data-latex="A_\infty"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g><g data-mml-node="mi" transform="translate(783,-150) scale(0.707)" data-latex="infty"><path data-c="221E" d="M749-11C807-11 855 13 892 60 926 104 943 156 943 216 943 275 926 327 893 371 856 418 809 442 752 442 684 442 625 416 576 364 547 332 524 303 507 278 464 329 435 361 421 373 367 419 310 442 250 442 192 442 144 418 107 371 73 327 56 275 56 215 56 156 73 104 106 60 143 13 190-11 247-11 315-11 374 15 423 67 452 99 475 128 492 153 535 102 564 70 578 58 632 12 689-11 749-11M913 216C913 188 911 168 908 156 903 137 890 117 869 94 840 61 805 44 765 44 722 44 680 67 637 113 592 168 559 209 538 237 601 348 675 403 759 403 852 403 913 314 913 216M86 215C86 260 100 299 128 334 156 369 191 387 234 387 277 387 319 364 362 318 407 263 440 222 461 194 398 83 324 28 240 28 147 28 86 117 86 215Z"></path></g></g></g></g></svg></mjx-container> - 环和<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.357ex" xmlns="http://www.w3.org/2000/svg" width="3.457ex" height="1.895ex" role="img" focusable="false" viewBox="0 -680 1528.1 837.8"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="E_\infty"><g data-mml-node="msub" data-latex="E_\infty"><g data-mml-node="mi" data-latex="E"><path data-c="1D438" d="M199 656C199 646 210 641 231 641 271 641 291 637 291 628 291 625 289 618 286 605L156 82C151 60 141 47 126 42 119 40 101 39 70 39 48 39 38 37 38 16 38 5 49 0 70 0L583 0C607 0 609 2 617 19L657 109C684 171 703 217 714 248 712 257 709 263 698 263 691 263 685 258 681 249 656 190 632 145 608 113 568 59 509 39 415 39L270 39C260 39 253 39 249 40 243 40 240 42 240 45 240 47 242 54 245 67L311 334 406 334C453 334 478 328 483 315 485 310 486 303 486 293 486 280 484 264 479 245 477 240 476 236 476 233 476 223 482 218 493 218 502 218 508 226 512 241L568 473C568 483 563 488 552 488 544 488 538 481 535 467 525 428 512 403 495 391 478 379 450 373 409 373L321 373 379 606C387 640 387 641 429 641L568 641C619 641 654 636 673 626 701 611 715 581 715 536 715 519 714 501 711 484 711 482 711 479 710 475L710 466C710 455 715 450 725 450 735 450 741 459 743 478L763 649C766 677 759 680 732 680L233 680C210 680 199 679 199 656Z"></path></g><g data-mml-node="mi" transform="translate(771,-150) scale(0.707)" data-latex="infty"><path data-c="221E" d="M749-11C807-11 855 13 892 60 926 104 943 156 943 216 943 275 926 327 893 371 856 418 809 442 752 442 684 442 625 416 576 364 547 332 524 303 507 278 464 329 435 361 421 373 367 419 310 442 250 442 192 442 144 418 107 371 73 327 56 275 56 215 56 156 73 104 106 60 143 13 190-11 247-11 315-11 374 15 423 67 452 99 475 128 492 153 535 102 564 70 578 58 632 12 689-11 749-11M913 216C913 188 911 168 908 156 903 137 890 117 869 94 840 61 805 44 765 44 722 44 680 67 637 113 592 168 559 209 538 237 601 348 675 403 759 403 852 403 913 314 913 216M86 215C86 260 100 299 128 334 156 369 191 387 234 387 277 387 319 364 362 318 407 263 440 222 461 194 398 83 324 28 240 28 147 28 86 117 86 215Z"></path></g></g></g></g></svg></mjx-container> - 环)。</p><p>这个理论和当代纯数学中极度活跃的前沿领域 —— <strong>稳定同伦理论（stable homotopy theory）</strong> 和 <strong>导出代数几何（derived algebraic geometry）</strong> 等理论不谋而合。</p></div></div><div class="footer"></div><div class="article-meta" id="bottom"><div class="new-meta-box"></div></div><div class="prev-next"><a class="prev" href="/notes/person/5"><p class="title"><i class="fa-solid fa-chevron-left" aria-hidden="true"></i>Wall英国数学家C.T.C. 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data-srcset="https://static.mhuig.top/npm/imbox@0.0.15/c/120.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="Spanier埃德温·斯潘尼尔的数学人生与多领域贡献"> <span class="title">Spanier埃德温·斯潘尼尔的数学人生与多领域贡献</span></a></div></div></article><article class="post white-box shadow floatable blur" id="comments"><span hidden=""><meta itemprop="discussionUrl" content="/notes/person/6#comments"></span><p ct=""><i class="fa-duotone fa-comments"></i> 留言区</p><div id="layoutHelper-comments"></div></article></div><aside id="l_side" itemscope="" itemtype="http://schema.org/WPSideBar"><section class="widget text desktop mobile pjax"><header><a href="/notes/"><i class="fa-duotone fa-book fa-fw" aria-hidden="true"></i> <span class="name">Notes</span></a></header><div class="content"></div></section><section class="widget list group desktop mobile pjax"><header><i class="fa-duotone fa-p fa-fw" aria-hidden="true"></i> <span class="name">History of mathematics</span></header><div class="content"><ul class="list entry navigation"><li><a class="flat-box" title="/notes/person/" href="/notes/person/" active-action="action-notesperson"><div class="name"> Welcome</div></a></li><li><a class="flat-box" title="/notes/person/1" href="/notes/person/1" active-action="action-notesperson1"><div class="name"> Hilbert</div></a></li><li><a class="flat-box" title="/notes/person/2" href="/notes/person/2" active-action="action-notesperson2"><div class="name"> Thompson</div></a></li><li><a class="flat-box" title="/notes/person/3" href="/notes/person/3" active-action="action-notesperson3"><div class="name"> Conway</div></a></li><li><a class="flat-box" title="/notes/person/4" href="/notes/person/4" active-action="action-notesperson4"><div class="name"> Borcherds</div></a></li><li><a class="flat-box" title="/notes/person/5" href="/notes/person/5" active-action="action-notesperson5"><div class="name"> Wall</div></a></li><li><a class="flat-box" 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