{"id":27307,"date":"2021-06-29T19:27:32","date_gmt":"2021-06-29T11:27:32","guid":{"rendered":"https:\/\/swarma.org\/?p=27307"},"modified":"2021-06-29T19:27:32","modified_gmt":"2021-06-29T11:27:32","slug":"%e6%9e%84%e5%bb%ba%e6%95%b0%e5%ad%a6%e5%92%8c%e7%89%a9%e7%90%86%e5%9f%ba%e7%a1%80%e7%9a%84%e8%8c%83%e7%95%b4%e8%ae%ba%ef%bc%9a%e7%94%a8%e3%80%8c%e7%ad%89%e4%bb%b7%e3%80%8d%e5%8f%96%e4%bb%a3%e3%80%8c","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=27307","title":{"rendered":"\u6784\u5efa\u6570\u5b66\u548c\u7269\u7406\u57fa\u7840\u7684\u8303\u7574\u8bba\uff1a\u7528\u300c\u7b49\u4ef7\u300d\u53d6\u4ee3\u300c\u76f8\u7b49\u300d"},"content":{"rendered":"
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\u7b49\u53f7\u662f\u6570\u5b66\u7684\u57fa\u77f3\uff0c\u6570\u5b66\u4e2d\u7684\u76f8\u7b49\uff08equality\uff09\u4f3c\u4e4e\u662f\u6700\u6ca1\u4e89\u8bae\u7684\u6982\u5ff5\u3002\u4f46\u8d8a\u6765\u8d8a\u591a\u7684\u6570\u5b66\u5bb6\u5f00\u59cb\u8ba4\u4e3a\uff0c\u7b49\u53f7\u662f\u6570\u5b66\u7684\u539f\u521d\u9519\u8bef\uff0c\u4ed6\u4eec\u60f3\u8981\u7528\u7b49\u4ef7\uff08equivalence\uff09\u7684\u8bed\u8a00\u91cd\u65b0\u8868\u8ff0\u6570\u5b66\uff0c\u4e0d\u662f\u5173\u6ce8\u63cf\u8ff0\u5bf9\u8c61\u7684\u5177\u4f53\u65b9\u5f0f\uff0c\u800c\u662f\u5c06\u5bf9\u8c61\u76f8\u4e92\u5173\u8054\u7684\u5404\u79cd\u4e0d\u540c\u65b9\u5f0f\u8003\u8651\u5728\u5185\u3002\u8fd9\u79cd\u5173\u6ce8\u7b49\u4ef7\u6027\u7684\u6570\u5b66\u7406\u8bba\u5c31\u662f\u6240\u8c13\u7684\u8303\u7574\u8bba\uff08category theory\uff09\u3002<\/span><\/strong><\/p>\n


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\u6570\u5b66\u5bb6 Jacob Lurie \u5148\u540e\u5199\u4f5c\u4e86944\u9875\u7684\u300a\u9ad8\u9636\u8303\u7574\u8bba\u300b\u548c1553\u9875\u7684\u300a\u9ad8\u9636\u4ee3\u6570\u300b\u6765\u9610\u8ff0\u8303\u7574\u8bba\u7684\u601d\u60f3\uff0c\u8fd9\u4e24\u672c\u7a0b\u7891\u5f0f\u8457\u4f5c\u7684\u5f71\u54cd\u88ab\u8ba4\u4e3a\u582a\u4e0e\u683c\u7f57\u6ed5\u8fea\u514b\u7684\u4ee3\u6570\u51e0\u4f55\u9769\u547d\u76f8\u63d0\u5e76\u8bba\u3002\u4f46\u65b0\u601d\u60f3\u7684\u63d0\u51fa\u4e5f\u5e26\u6765\u4e86\u5de8\u5927\u7684\u6311\u6218\uff1a\u6570\u5b66\u754c\u8981\u5982\u4f55\u5438\u6536\u8fd9\u4e9b\u65b0\u77e5\u8bc6\uff1f\u5f53\u6570\u5b66\u88ab\u91cd\u65b0\u4e66\u5199\u65f6\uff0c\u6570\u5b66\u5bb6\u7fa4\u4f53\u8981\u5982\u4f55\u5e94\u5bf9\uff1f\u77e5\u8bc6\u7684\u771f\u6b63\u76ee\u7684\u5230\u5e95\u662f\u4ec0\u4e48\uff1f<\/span><\/strong><\/p>\n


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\u4e8b\u5b9e\u4e0a\uff0c\u8303\u7574\u8bba\u9664\u4e86\u4f5c\u4e3a\u4e00\u79cd\u6781\u7aef\u62bd\u8c61\u7684\u6570\u5b66\u7406\u8bba\u4e4b\u5916\uff0c\u4e5f\u5df2\u7ecf\u5e94\u7528\u5230\u4e86\u7269\u7406\u5b66\u9886\u57df\u6765\u63cf\u5199\u591a\u4f53\u91cf\u5b50\u7ea0\u7f20\uff08\u4e5f\u5c31\u662f\u62d3\u6251\u5e8f\uff09\u8fd9\u79cd\u5168\u65b0\u7684\u81ea\u7136\u73b0\u8c61\u3002\u5728\u540e\u8bb0\u4e2d\uff0c\u300c\u8fd4\u6734\u300d\u603b\u7f16\u6587\u5c0f\u521a\u6559\u6388\u3001\u6df1\u5733\u91cf\u5b50\u79d1\u5b66\u4e0e\u5de5\u7a0b\u7814\u7a76\u9662\u5b54\u826f\u7814\u7a76\u5458\u4ecb\u7ecd\u4e86\u6570\u5b66\u548c\u7269\u7406\u4e4b\u95f4\u8fd9\u79cd\u6df1\u523b\u7684\u5173\u8054\uff0c\u4ed6\u4eec\u8ba4\u4e3a\uff0c\u8fd9\u4e2a\u65f6\u4ee3\u662f\u6570\u5b66\u548c\u7269\u7406\u878d\u5408\u7684\u9ec4\u91d1\u65f6\u4ee3\u3002<\/span><\/strong><\/p>\n


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\u4e3a\u4e86\u8ba9\u5927\u5bb6\u4e86\u89e3\u8303\u7574\u8bba\u8fd9\u6837\u4e00\u95e8\u73b0\u4ee3\u6570\u5b66\u8bed\u8a00\uff0c\u514b\u670d\u4f20\u7edf\u5b66\u4e60\u8303\u7574\u8bba\u62bd\u8c61\u548c\u5bf9\u524d\u7f6e\u77e5\u8bc6\u7684\u969c\u788d\uff0c\u96c6\u667a\u5b66\u56ed\u7279\u9080\u4e00\u4f4d\u6b63\u5728\u5c1d\u8bd5\u6559\u4e2d\u5b66\u751f\u8303\u7574\u8bba\u7684J-CAT\u732b\u5708\u8001\u5e08\u5f00\u8bfe\uff0c\u7b79\u5212\u4e86\u201c\u96c6\u667a\u8303\u7574\u8bba\u5165\u95e8\u7cfb\u5217\u8bfe\u7a0b<\/a>\u201d\u3002\u5e0c\u671b\u8fd9\u95e8\u8bfe\u7a0b\u53ef\u4ee5\u5e2e\u52a9\u5927\u5bb6\u7834\u9664\u95e8\u69db\uff0c\u987a\u5229\u5165\u95e8\u3002<\/strong><\/p>\n


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\u6b64\u7cfb\u5217\u8bfe\u7a0b\u4e3a\u5468\u66f4\u8bfe\u7a0b\uff0c\u6bcf\u5468\u65e5\u4e2d\u534812\u70b9\u66f4\u65b0\u3002\u6b22\u8fce\u5bf9\u8303\u7574\u8bba\u611f\u5174\u8da3\u7684\u670b\u53cb\u62a5\u540d\u52a0\u5165\u8bfe\u7a0b\u3002\u8be6\u60c5\u8bf7\u89c1\u6587\u672b\u5185\u5bb9\u3002<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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Kevin Hartnett | \u4f5c\u8005<\/strong><\/span><\/p>\n

\u8fd4\u6734\uff08ID\uff1afanpu2019\uff09 | \u6765\u6e90<\/strong><\/span><\/p>\n


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\u7b49\u53f7\u662f\u6570\u5b66\u7684\u57fa\u77f3\u3002\u5b83\u4f3c\u4e4e\u6784\u6210\u4e86\u4e00\u4e2a\u5b8c\u5168\u57fa\u672c\u548c\u65e0\u53ef\u4e89\u8bae\u7684\u547d\u9898\uff1a\u8fd9\u4e9b\u4e1c\u897f\u662f\u5b8c\u5168\u4e00\u6837\u7684\u3002<\/span>
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\u4f46\u8d8a\u6765\u8d8a\u591a\u7684\u6570\u5b66\u5bb6\u8ba4\u4e3a\u7b49\u53f7\u662f\u6570\u5b66\u7684\u539f\u521d\u9519\u8bef\u3002\u4ed6\u4eec\u5c06\u5176\u89c6\u4e3a\u4e00\u79cd\u865a\u9970\uff0c\u5b83\u63a9\u76d6\u4e86\u91cf\u7684\u76f8\u5173\u65b9\u5f0f\u4e2d\u91cd\u8981\u7684\u590d\u6742\u6027\u2014\u2014\u8fd9\u4e9b\u590d\u6742\u6027\u53ef\u4ee5\u63ed\u793a\u5927\u91cf\u95ee\u9898\u7684\u7b54\u6848\u3002\u4ed6\u4eec\u60f3\u7528\u7b49\u4ef7\u8fd9\u79cd\u66f4\u5bbd\u677e\u7684\u8bed\u8a00\u6765\u91cd\u65b0\u8868\u8ff0\u6570\u5b66\u3002<\/span><\/p>\n


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\u675c\u514b\u5927\u5b66\u7684\u4e54\u7eb3\u68ee\u00b7\u574e\u8d1d\u5c14<\/span>\uff08Jonathan Campbell\uff09<\/span>\u8868\u793a\uff1a\u201c\u6211\u4eec\u4e00\u76f4\u7528\u7684\u662f\u76f8\u7b49<\/span>\uff08equality\uff09<\/span>\u7684\u6982\u5ff5\uff0c\u5176\u5b9e\u5e94\u5f53\u662f\u7b49\u4ef7<\/span>\uff08equivalence\uff09<\/span>\u3002\u201d<\/span><\/p>\n


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\u8fd9\u4e9b\u6570\u5b66\u5bb6\u4e2d\u6700\u6770\u51fa\u7684\u662f\u96c5\u5404\u5e03\u00b7\u5362\u91cc<\/span>\uff08Jacob Lurie\uff09<\/span>\u3002\u4eca\u5e747\u6708\uff0c41\u5c81\u7684\u5362\u91cc\u79bb\u5f00\u4e86\u4ed6\u5728\u54c8\u4f5b\u5927\u5b66\u7684\u7ec8\u8eab\u804c\u4f4d\uff0c\u524d\u5f80\u65b0\u6cfd\u897f\u5dde\u666e\u6797\u65af\u987f\u9ad8\u7b49\u7814\u7a76\u9662\u4efb\u6559\uff0c\u90a3\u91cc\u4e91\u96c6\u4e86\u4e16\u754c\u4e0a\u8bb8\u591a\u6700\u6709\u58f0\u671b\u7684\u6570\u5b66\u5bb6\u3002<\/span><\/p>\n


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\u5362\u91cc\u7684\u601d\u60f3\u5728\u4efb\u4f55\u9886\u57df\u90fd\u662f\u7a7a\u524d\u7edd\u540e\u7684\u3002\u4ed6\u7528\u539a\u8fbe\u5343\u9875\u7684\u4e13\u4e1a\u8457\u4f5c\uff0c\u901a\u8fc7\u8d85\u8d8a\u7b49\u53f7\uff0c\u6784\u5efa\u4e86\u4e00\u4e2a\u660e\u663e\u4e0d\u540c\u7684\u65b9\u5f0f\u6765\u7406\u89e3\u4e00\u4e9b\u6700\u91cd\u8981\u7684\u6570\u5b66\u6982\u5ff5\u3002\u5362\u91cc\u7684\u5bfc\u5e08\u3001\u54c8\u4f5b\u5927\u5b66\u6570\u5b66\u5bb6\u8fc8\u514b\u5c14\u00b7\u970d\u666e\u91d1\u65af<\/span>\uff08Michael Hopkins\uff09<\/span>\u8bf4\uff1a\u201c\u6211\u60f3\u4ed6\u8ba4\u4e3a\u8fd9\u624d\u662f\u601d\u8003\u6570\u5b66\u7684\u6b63\u786e\u65b9\u5f0f\u3002\u201d<\/span><\/p>\n


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\u5362\u91cc\u57282009\u5e74\u51fa\u7248\u4e86\u4ed6\u7684\u7b2c\u4e00\u672c\u4e66\u300a\u9ad8\u9636\u8303\u7574\u8bba\u300b<\/span>\uff08Higher Topos Theory\uff09<\/span>\u3002\u8fd9\u672c944\u9875\u7684\u4e66\u5c31\u50cf\u4e00\u672c\u624b\u518c\uff0c\u6559\u4f60\u5982\u4f55\u7528\u65b0\u7684\u201c\u65e0\u7a77\u8303\u7574<\/span>\uff08infinity categories\uff09<\/span>\u201d\u7684\u8bed\u8a00\u6765\u89e3\u91ca\u5df2\u7ecf\u5efa\u7acb\u8d77\u6765\u7684\u6570\u5b66\u9886\u57df\u3002\u5728\u90a3\u4e4b\u540e\u7684\u51e0\u5e74\u91cc\uff0c\u5362\u91cc\u7684\u601d\u60f3\u5f71\u54cd\u5230\u8d8a\u6765\u8d8a\u5e7f\u6cdb\u7684\u6570\u5b66\u9886\u57df\u3002\u8bb8\u591a\u6570\u5b66\u5bb6\u8ba4\u4e3a\u5b83\u4eec\u5bf9\u6570\u5b66\u7684\u672a\u6765\u662f\u4e0d\u53ef\u6216\u7f3a\u7684\u3002\u897f\u5317\u5927\u5b66\u7684\u7ea6\u7ff0\u00b7\u5f17\u6717\u897f\u65af<\/span>\uff08John Francis\uff09<\/span>\u8bf4\uff1a\u201c\u4e00\u65e6\u5b66\u4f1a\u4e86\u65e0\u7a77\u8303\u7574\uff0c\u5c31\u6ca1\u6709\u4eba\u4f1a\u56de\u5934\u3002\u201d<\/span><\/p>\n


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IAS\u6570\u5b66\u5bb6\u96c5\u5404\u5e03\u00b7\u5362\u91cc\u57282014\u5e74\u83b7\u5f97\u4e86300\u4e07\u7f8e\u5143\u7684\u6570\u5b66\u7a81\u7834\u5956\u3002| \u56fe\u7247\u6765\u6e90\uff1a\u9ea6\u514b\u963f\u745f\u57fa\u91d1\u4f1a<\/span><\/p>\n


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\u7136\u800c\uff0c\u65e0\u7a77\u8303\u7574\u7684\u6269\u5c55\u4e5f\u63ed\u793a\u51fa\uff0c\u50cf\u6570\u5b66\u8fd9\u6837\u7684\u53e4\u8001\u9886\u57df\u4e00\u65e6\u8bd5\u56fe\u5438\u6536\u67d0\u4e2a\u91cd\u5927\u7684\u65b0\u601d\u60f3\uff0c\u5c24\u5176\u662f\u5f53\u8fd9\u79cd\u601d\u60f3\u4f1a\u6311\u6218\u5176\u6700\u91cd\u8981\u6982\u5ff5\u7684\u610f\u4e49\u65f6\uff0c\u5b83\u5c06\u4e0d\u5f97\u4e0d\u7ecf\u5386\u6210\u957f\u7684\u75db\u82e6\u3002\u7231\u4e01\u5821\u5927\u5b66\u7684\u514b\u62c9\u514b\u00b7\u5df4\u7ef4\u514b<\/span>\uff08Clark Barwick\uff09<\/span>\u8bf4\uff1a\u201c\u6570\u5b66\u754c\u7684\u4fdd\u5b88\u529b\u91cf\u5f88\u5f3a\u3002\u5982\u679c\u4e0d\u80fd\u7ed9\u51fa\u4ee4\u4eba\u4fe1\u670d\u7684\u7406\u7531\uff0c\u5c31\u4e0d\u8981\u6307\u671b\u4efb\u4f55\u4e00\u7fa4\u6570\u5b66\u5bb6\u4f1a\u6beb\u4e0d\u8fdf\u7591\u5730\u8fc5\u901f\u63a5\u53d7\u4efb\u4f55\u65b0\u5de5\u5177\u3002\u201d<\/span><\/p>\n


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\u5c3d\u7ba1\u8bb8\u591a\u6570\u5b66\u5bb6\u5df2\u7ecf\u63a5\u53d7\u4e86\u65e0\u7a77\u8303\u7574\uff0c\u4f46\u662f\u5f88\u5c11\u6709\u4eba\u5b8c\u6574\u9605\u8bfb\u8fc7\u5362\u91cc\u9ad8\u5ea6\u62bd\u8c61\u7684\u957f\u7bc7\u4e13\u8457\u3002\u56e0\u6b64\uff0c\u4e00\u4e9b\u57fa\u4e8e\u4ed6\u7684\u601d\u60f3\u7684\u5de5\u4f5c\u5e76\u4e0d\u50cf\u6570\u5b66\u4e2d\u901a\u5e38\u90a3\u6837\u4e25\u8c28\u3002<\/span><\/p>\n


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\u5eb7\u5948\u5c14\u5927\u5b66\u6570\u5b66\u5bb6\u8335\u5a1c\u00b7\u624e\u54c8\u91cc\u7ef4\u5947<\/span>\uff08Inna Zakharevich\uff09<\/span>\u8bf4\uff1a\u201c\u6211\u542c\u5230\u4eba\u8bf4\uff0c\u2018\u5728\u5362\u91cc\u7684\u4e66\u91cc\u8bb2\u8fc7\u3002\u2019\u6211\u8bf4\uff0c\u2018\u771f\u7684\u5417\uff1f\u4f60\u5f15\u7528\u7684\u662f8000\u9875\u7684\u6587\u732e\u3002\u8fd9\u4e0d\u662f\u5f15\u7528\uff0c\u8fd9\u662f\u62b1\u5927\u817f\u3002\u2019\u201d<\/span><\/p>\n


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\u6570\u5b66\u5bb6\u4eec\u4ecd\u7136\u5728\u52aa\u529b\u7406\u89e3\u5362\u91cc\u7684\u601d\u60f3\u7684\u91cd\u8981\u6027\u548c\u4ecb\u7ecd\u5b83\u4eec\u7684\u72ec\u7279\u65b9\u5f0f\u3002\u4ed6\u4eec\u8fd8\u5728\u63d0\u70bc\u548c\u91cd\u65b0\u5305\u88c5\u4ed6\u5bf9\u65e0\u7a77\u8303\u7574\u7684\u8868\u73b0\u65b9\u5f0f\uff0c\u4ee5\u4fbf\u8ba9\u66f4\u591a\u7684\u6570\u5b66\u5bb6\u80fd\u7406\u89e3\u5b83\u4eec\u3002\u5728\u67d0\u79cd\u610f\u4e49\u4e0a\uff0c\u4ed6\u4eec\u6b63\u5728\u4ece\u4e8b\u4efb\u4f55\u9769\u547d\u4e4b\u540e\u5fc5\u987b\u8fdb\u884c\u7684\u6cbb\u7406\u5de5\u4f5c\uff0c\u5c06\u53d8\u9769\u6027\u6587\u672c\u8f6c\u5316\u4e3a\u65e5\u5e38\u6cd5\u5f8b\u3002\u901a\u8fc7\u8fd9\u6837\u505a\uff0c\u4ed6\u4eec\u5c06\u6570\u5b66\u7684\u672a\u6765\u5efa\u7acb\u5728\u7b49\u4ef7\u7684\u57fa\u7840\u4e0a\uff0c\u800c\u4e0d\u518d\u662f\u5728\u76f8\u7b49\u7684\u57fa\u7840\u4e0a\u3002<\/span><\/p>\n


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\u70b9\u7b49\u4ef7\u3002\u5982\u679c\u4e24\u70b9\u4e4b\u95f4\u81f3\u5c11\u6709\u4e00\u6761\u8def\u5f84\u76f8\u8fde\uff0c\u5219\u4e24\u70b9\u540c\u4f26\u7b49\u4ef7\u3002| \u56fe\u7247\u6765\u6e90\uff1aQuanta Magazine<\/span><\/p>\n


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\u4f46\u4ec5\u4ec5\u8bf4\u4e24\u70b9\u662f\u7531\u8bb8\u591a\u7b49\u540c\u7684\u8def\u5f84\u8fde\u63a5\u5230\u4e00\u8d77\u8fd8\u4e0d\u591f\u3002\u4f60\u8fd8\u9700\u8981\u8003\u8651\u6240\u6709\u8fd9\u4e9b\u8def\u5f84\u4e4b\u95f4\u7684\u7b49\u4ef7\u6027\u3002\u56e0\u6b64\uff0c\u9664\u4e86\u8981\u95ee\u4e24\u70b9\u662f\u5426\u7b49\u4ef7\u4e4b\u5916\uff0c\u4f60\u8fd8\u8981\u95ee\uff0c\u5728\u540c\u4e00\u5bf9\u70b9\u4e0a\u5f00\u59cb\u548c\u7ed3\u675f\u7684\u4e24\u6761\u8def\u5f84\u662f\u5426\u7b49\u4ef7\u2014\u2014\u662f\u5426\u6709\u4e00\u6761\u8def\u5f84\u5173\u8054\u8fd9\u4e24\u6761\u8def\u5f84\u3002\u4e24\u6761\u8def\u5f84\u4e4b\u95f4\u7684\u8fd9\u6761\u8def\u5f84\u7684\u5f62\u72b6\u4e3a\u4e00\u4e2a\u76d8\uff0c\u76d8\u7684\u8fb9\u754c\u5c31\u662f\u8fd9\u4e24\u6761\u8def\u5f84\u3002<\/span><\/p>\n


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\u8def\u5f84\u7b49\u4ef7\u3002\u5982\u679c\u81f3\u5c11\u6709\u4e00\u4e2a\u76d8\u8fde\u63a5\u4e24\u6761\u8def\u5f84\uff0c\u5219\u4e24\u6761\u8def\u5f84\u540c\u4f26\u7b49\u4ef7\u3002| \u56fe\u7247\u6765\u6e90\uff1aQuanta Magazine<\/span><\/p>\n


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\u4f60\u53ef\u4ee5\u7ee7\u7eed\u63a8\u8fdb\u3002\u5982\u679c\u4e24\u4e2a\u76d8\u4e4b\u95f4\u6709\u4e00\u6761\u8def\u5f84\uff0c\u90a3\u4e48\u8fd9\u4e24\u4e2a\u76d8\u5c31\u662f\u7b49\u4ef7\u7684\uff0c\u800c\u8fd9\u6761\u8def\u5f84\u7684\u5f62\u72b6\u5c06\u662f\u4e09\u7ef4\u5bf9\u8c61\u3002\u8fd9\u4e9b\u4e09\u7ef4\u5bf9\u8c61\u53c8\u53ef\u80fd\u901a\u8fc7\u56db\u7ef4\u8def\u5f84\u76f8\u4e92\u5173\u8054\uff08<\/span>\u4e24\u4e2a\u5bf9\u8c61\u4e4b\u95f4\u7684\u8def\u5f84\u603b\u662f\u6bd4\u5bf9\u8c61\u672c\u8eab\u591a\u4e00\u4e2a\u7ef4\u5ea6\uff09<\/span>\u3002<\/span><\/p>\n


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\u6700\u7ec8\uff0c\u4f60\u5c06\u5efa\u7acb\u4e00\u4e2a\u7b49\u4ef7\u5173\u7cfb\u7684\u65e0\u7a77\u5854\u3002\u5982\u679c\u901a\u76d8\u8003\u8651\u6574\u4e2a\u5854\uff0c\u4f60\u5c31\u53ef\u4ee5\u5bf9\u4f60\u6240\u9009\u62e9\u7684\u7528\u7403\u9762\u4e0a\u7684\u70b9\u8868\u793a\u7684\u4efb\u4f55\u5bf9\u8c61\u5f62\u6210\u5168\u9762\u7684\u8ba4\u8bc6\u3002<\/span><\/p>\n


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\u5fb7\u5dde\u5927\u5b66\u5965\u65af\u6c40\u5206\u6821\u7684\u5927\u536b\u00b7\u672c-\u5179\u7ef4<\/span>\uff08David Ben-Zvi\uff09<\/span>\u8bf4\uff1a\u201c\u5b83\u53ea\u662f\u4e00\u4e2a\u7403\uff0c\u4f46\u4e8b\u5b9e\u8bc1\u660e\uff0c\u8981\u7406\u89e3\u7403\u7684\u5f62\u72b6\uff0c\u4ece\u67d0\u79cd\u610f\u4e49\u4e0a\u8bf4\uff0c\u4f60\u9700\u8981\u5230\u65e0\u7a77\u8fdc\u5904\u53bb\u3002\u201d<\/span><\/p>\n


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\u572820\u4e16\u7eaa\u7684\u6700\u540e\u51e0\u5341\u5e74\u91cc\uff0c\u8bb8\u591a\u6570\u5b66\u5bb6\u81f4\u529b\u4e8e\u4e00\u4e2a\u201c\u65e0\u7a77\u8303\u7574\u201d\u7406\u8bba\u2014\u2014\u8fd9\u4e2a\u7406\u8bba\u53ef\u4ee5\u7814\u7a76\u7b49\u4ef7\u5173\u7cfb\u7684\u65e0\u7a77\u5854\u3002\u6709\u51e0\u4e2a\u4eba\u53d6\u5f97\u4e86\u5b9e\u8d28\u6027\u8fdb\u5c55\u3002\u4f46\u53ea\u6709\u4e00\u4e2a\u4eba\u6210\u529f\u4e86\u3002<\/span><\/p>\n


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2.\u91cd\u5199\u6570\u5b66<\/strong><\/strong><\/span><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

\u96c5\u5404\u5e03\u00b7\u5362\u91cc\u5173\u4e8e\u65e0\u7a77\u8303\u7574\u7406\u8bba\u7684\u7b2c\u4e00\u7bc7\u8bba\u6587\u5e76\u4e0d\u6210\u529f\u30022003\u5e746\u67085\u65e5\uff0c\u8fd9\u4f4d25\u5c81\u7684\u5e74\u8f7b\u4eba\u5728\u79d1\u5b66\u9884\u5370\u672c\u7f51\u7ad9 arxiv.org \u4e0a\u53d1\u5e03\u4e86\u4e00\u4efd60\u9875\u7684\u8bba\u6587\uff0c\u9898\u4e3a\u300a\u8bba\u65e0\u7a77\u8303\u7574\u300b<\/span>\uff08On Infinity Topoi\uff09<\/span>\u3002\u5728\u6587\u4e2d\u4ed6\u5f00\u59cb\u52fe\u52d2\u4e00\u4e9b\u89c4\u5219\uff0c\u6570\u5b66\u5bb6\u4eec\u53ef\u4ee5\u7528\u8fd9\u4e9b\u89c4\u5219\u7814\u7a76\u65e0\u7a77\u8303\u7574\u3002<\/span><\/p>\n


<\/p>\n

\u7b2c\u4e00\u7bc7\u8bba\u6587\u6ca1\u6709\u5f97\u5230\u666e\u904d\u597d\u8bc4\u3002\u8bfb\u5b8c\u6587\u7ae0\u540e\u4e0d\u4e45\uff0c\u829d\u52a0\u54e5\u5927\u5b66\u6570\u5b66\u5bb6\u5f7c\u5f97\u00b7\u6885<\/span>\uff08Peter May\uff09<\/span>\u7ed9\u5362\u91cc\u7684\u5bfc\u5e08\u8fc8\u514b\u5c14\u00b7\u970d\u666e\u91d1\u65af\u53d1\u4e86\u5c01\u7535\u5b50\u90ae\u4ef6\uff0c\u8bf4\u5362\u91cc\u7684\u8bba\u6587\u6709\u4e00\u4e9b\u6709\u8da3\u7684\u60f3\u6cd5\uff0c\u4f46\u611f\u89c9\u5f88\u4e0d\u5b8c\u5584\uff0c\u9700\u8981\u66f4\u4e25\u683c\u3002\u6885\u8bf4\uff1a\u201c\u6211\u5411\u8fc8\u514b\u5c14\u8868\u8fbe\u4e86\u6211\u4eec\u7684\u4fdd\u7559\u610f\u89c1\uff0c\u8fc8\u514b\u5c14\u8f6c\u8fbe\u7ed9\u4e86\u96c5\u5404\u5e03\u3002\u201d<\/span><\/p>\n


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\u4e0d\u6e05\u695a\u5362\u91cc\u662f\u5426\u66fe\u628a\u6885\u7684\u90ae\u4ef6\u89c6\u4e3a\u4e00\u79cd\u6311\u6218\uff0c\u6216\u8005\u4ed6\u65e9\u5c31\u8ba1\u5212\u597d\u4e86\u4e0b\u4e00\u6b65\u7684\u884c\u52a8\u3002<\/span>\uff08\u5362\u91cc\u591a\u6b21\u62d2\u7edd\u4e86\u5c31\u6b64\u4e8b\u63a5\u53d7\u91c7\u8bbf\u7684\u8bf7\u6c42\u3002\uff09<\/span>\u6211\u4eec\u53ea\u77e5\u9053\uff0c\u5728\u53d7\u5230\u6279\u8bc4\u540e\uff0c\u5362\u91cc\u8fdb\u5165\u4e86\u4e00\u4e2a\u65fa\u76db\u7684\u591a\u4ea7\u671f\uff0c\u8fd9\u6bb5\u65f6\u671f\u5df2\u7ecf\u6210\u4e3a\u4f20\u5947\u3002<\/span><\/p>\n


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\u6885\u8bf4\uff1a\u201c\u6211\u65e0\u6cd5\u8fdb\u5165\u96c5\u5404\u5e03\u7684\u8111\u5b50\u91cc\uff0c\u6211\u4e0d\u80fd\u786e\u5207\u5730\u8bf4\u51fa\u4ed6\u5f53\u65f6\u5728\u60f3\u4ec0\u4e48\u3002\u4f46\u6beb\u65e0\u7591\u95ee\uff0c\u6211\u4eec\u6279\u8bc4\u7684\u6587\u7a3f\u4e0e\u6700\u7ec8\u7248\u672c\u4e4b\u95f4\u5b58\u5728\u5de8\u5927\u5dee\u5f02\uff0c\u540e\u8005\u5b8c\u5168\u662f\u5728\u66f4\u9ad8\u7684\u6570\u5b66\u5c42\u9762\u4e0a\u3002\u201d<\/span><\/p>\n


<\/p>\n

2006\u5e74\uff0c\u5362\u91cc\u5728 arxiv.org\u4e0a \u53d1\u5e03\u4e86\u300a\u9ad8\u9636\u8303\u7574\u8bba\u300b\u7684\u4e66\u7a3f\u3002\u5728\u8fd9\u9879\u91cc\u7a0b\u7891\u5f0f\u7684\u6210\u5c31\u4e2d\uff0c\u4ed6\u7528\u4e00\u79cd\u65b0\u7684\u6570\u5b66\u57fa\u7840\uff0c\u57fa\u4e8e\u65e0\u7a77\u8303\u7574\u7684\u57fa\u7840\uff0c\u5efa\u7acb\u4e86\u53d6\u4ee3\u96c6\u5408\u8bba\u7684\u673a\u5236\u3002\u4f0a\u5229\u8bfa\u4f0a\u5927\u5b66\u5384\u5df4\u7eb3-\u9999\u69df\u5206\u6821\u6570\u5b66\u5bb6\u67e5\u5c14\u65af\u00b7\u745e\u6cfd\u514b<\/span>\uff08Charles Rezk\uff09<\/span>\u505a\u4e86\u5173\u4e8e\u65e0\u7a77\u8303\u7574\u8bba\u91cd\u8981\u7684\u65e9\u671f\u5de5\u4f5c\uff0c\u4ed6\u8bf4\uff1a\u201c\u4ed6\u7528\u4e0a\u5343\u9875\u7bc7\u5e45\u521b\u9020\u4e86\u6211\u4eec\u73b0\u5728\u90fd\u5728\u4f7f\u7528\u7684\u57fa\u7840\u673a\u5236\u3002\u6211\u60f3\u6211\u7528\u4e00\u8f88\u5b50\u90fd\u5199\u4e0d\u51fa\u300a\u9ad8\u9636\u8303\u7574\u8bba\u300b\uff0c\u4ed6\u7528\u4e24\u4e09\u5e74\u5c31\u5b8c\u6210\u4e86\u3002\u201d<\/span><\/p>\n


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\u7136\u540e\u57282011\u5e74\uff0c\u5362\u91cc\u53c8\u5199\u4e86\u4e00\u672c\u7bc7\u5e45\u66f4\u957f\u7684\u4e13\u8457\u3002\u5728\u4e66\u4e2d\uff0c\u4ed6\u91cd\u65b0\u53d1\u660e\u4e86\u4ee3\u6570\u3002<\/span><\/p>\n


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\u4ee3\u6570\u4e3a\u5904\u7406\u65b9\u7a0b\u5f0f\u63d0\u4f9b\u4e86\u4e00\u5957\u4f18\u96c5\u7684\u5f62\u5f0f\u89c4\u5219\u3002\u6570\u5b66\u5bb6\u4eec\u4e00\u76f4\u4f7f\u7528\u8fd9\u4e9b\u89c4\u5219\u6765\u8bc1\u660e\u65b0\u5b9a\u7406\u3002\u4f46\u662f\u4ee3\u6570\u662f\u5728\u56fa\u5b9a\u4e0d\u52a8\u7684\u7b49\u53f7\u5e73\u8861\u6728\u4e0a\u8868\u6f14\u4f53\u64cd\u3002\u5982\u679c\u4f60\u53bb\u6389\u8fd9\u4e9b\u5e73\u8861\u6728\uff0c\u7528\u66f4\u7cbe\u5de7\u7684\u7b49\u4ef7\u6982\u5ff5\u6765\u4ee3\u66ff\u5b83\u4eec\uff0c\u6709\u4e9b\u64cd\u4f5c\u5c31\u4f1a\u53d8\u5f97\u56f0\u96be\u5f97\u591a\u3002<\/span><\/p>\n


<\/p>\n

\u4ee5\u5c0f\u5b66\u6559\u6388\u7684\u7b2c\u4e00\u6761\u4ee3\u6570\u89c4\u5219\u7ed3\u5408\u5f8b\u4e3a\u4f8b\uff1a3\u4e2a\u62163\u4e2a\u4ee5\u4e0a\u6570\u5b57\u7684\u548c\u6216\u4e58\u79ef\u5e76\u4e0d\u53d6\u51b3\u4e8e\u8fd9\u4e9b\u6570\u5b57\u662f\u5982\u4f55\u5206\u7ec4\u7684\uff0c\u6bd4\u5982 2 \u00d7 (3 \u00d7 4) = (2 \u00d7 3) \u00d7 4\u3002<\/span><\/p>\n


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\u5982\u679c\u7528\u76f8\u7b49\u6982\u5ff5\uff0c\u8981\u8bc1\u660e\u7ed3\u5408\u5f8b\u5bf9\u4efb\u4f55\u5305\u542b3\u4e2a\u62163\u4e2a\u4ee5\u4e0a\u6570\u5b57\u7684\u5217\u8868\u90fd\u9002\u7528\u662f\u5f88\u5bb9\u6613\u7684\u3002\u4f46\u5982\u679c\u4f7f\u7528\u5f3a\u7b49\u4ef7\u6982\u5ff5\uff0c\u5c31\u4f1a\u5f88\u590d\u6742\u3002\u5982\u679c\u4f7f\u7528\u66f4\u7cbe\u5de7\u7684\u7b49\u4ef7\u6982\u5ff5\u65f6\uff0c\u5373\u4f7f\u662f\u50cf\u7ed3\u5408\u5f8b\u8fd9\u6837\u7684\u7b80\u5355\u89c4\u5219\u4e5f\u4f1a\u53d8\u5f97\u975e\u5e38\u68d8\u624b\u3002<\/span><\/p>\n


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\u7ed3\u5408\u5f8b\u56fe\u3002\u5728\u4ee3\u6570\u4e2d\uff0c\u7ed3\u5408\u5f8b\u544a\u8bc9\u6211\u4eec (a \u00d7 b) \u00d7 c = a \u00d7 (b \u00d7 c)\u3002\u800c\u4e00\u65e6\u5f15\u5165\u7b49\u4ef7\uff0c\u4ec5\u9760\u7ed3\u5408\u5f8b\u65e0\u6cd5\u4fdd\u8bc1\u6240\u6709\u7ec4\u5408\u90fd\u6709\u76f8\u540c\u7684\u4e58\u79ef\u3002\u8fd9\u4e2a\u6784\u56fe\u79f0\u4e3a\uff085\u5143\u7d20\uff09\u7ed3\u5408\u591a\u9762\u4f53\uff0c\u8868\u793a\u4e86\u7ec4\u5408\u4e4b\u95f4\u7684\u7b49\u4ef7\u5173\u7cfb\u3002\u56fe\u4e2d\u6bcf\u4e2a\u9876\u70b9\u8868\u793a\u4e00\u79cd\u7ec4\u5408\u3002\u5171\u8fb9\u548c\u5171\u9762\u7684\u7ec4\u5408\u6839\u636e\u7ed3\u5408\u5f8b\u76f8\u4e92\u7b49\u4ef7\u3002| \u56fe\u7247\u6765\u6e90\uff1aQuanta Magazine<\/span><\/p>\n


<\/p>\n

\u8499\u5927\u62ff\u5dde\u7acb\u5927\u5b66\u6570\u5b66\u5bb6\u6234\u7ef4\u00b7\u963f\u4e9a\u62c9<\/span>\uff08David Ayala\uff09<\/span>\u8bf4\uff1a\u201c\u8fd9\u4f7f\u95ee\u9898\u53d8\u5f97\u975e\u5e38\u590d\u6742\uff0c\u67d0\u79cd\u7a0b\u5ea6\u4e0a\u4f7f\u5f97\u6211\u4eec\u7406\u60f3\u7684\u65b0\u7248\u672c\u7684\u6570\u5b66\u4f3c\u4e4e\u4e0d\u73b0\u5b9e\u3002\u201d<\/span><\/p>\n


<\/p>\n

\u5728\u6700\u65b0\u7248\u957f\u8fbe1553\u9875\u7684\u300a\u9ad8\u9636\u4ee3\u6570\u300b<\/span>\uff08Higher Algebra\uff09<\/span>\u4e2d\uff0c\u5362\u91cc\u53d1\u5c55\u4e86\u65e0\u7a77\u8303\u7574\u7248\u672c\u7684\u7ed3\u5408\u5f8b\u2014\u2014\u4ee5\u53ca\u5176\u4ed6\u8bb8\u591a\u4ee3\u6570\u5b9a\u7406\uff0c\u5b83\u4eec\u5171\u540c\u5960\u5b9a\u4e86\u7b49\u4ef7\u6570\u5b66\u7684\u57fa\u7840\u3002<\/span><\/p>\n


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\u603b\u800c\u8a00\u4e4b\uff0c\u4ed6\u7684\u4e24\u672c\u4e66\u975e\u5e38\u9707\u64bc\uff0c\u662f\u5f15\u53d1\u79d1\u5b66\u9769\u547d\u7684\u90a3\u79cd\u8457\u4f5c\u3002\u91cc\u5c14\u8bf4\uff1a\u201c\u89c4\u6a21\u975e\u5e38\u5e9e\u5927\u3002\u8fd9\u4e2a\u6210\u5c31\u53ef\u4ee5\u4e0e\u683c\u7f57\u6ed5\u8fea\u514b<\/span>\uff08Grothendieck\uff09<\/span>\u7684\u4ee3\u6570\u51e0\u4f55\u9769\u547d\u76f8\u63d0\u5e76\u8bba\u3002\u201d<\/span><\/p>\n


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3.\u6d88\u5316\u4e00\u5934\u725b<\/strong><\/strong><\/span><\/p>\n<\/section>\n

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\u6570\u5b66\u5bb6\u4ee5\u601d\u7ef4\u6e05\u6670\u8457\u79f0\uff1a\u8bc1\u660e\u662f\u5426\u6b63\u786e\uff0c\u60f3\u6cd5\u662f\u5426\u6709\u6548\u3002\u4f46\u662f\u6570\u5b66\u5bb6\u4e5f\u662f\u4eba\u7c7b\uff0c\u4ed6\u4eec\u5bf9\u65b0\u60f3\u6cd5\u7684\u53cd\u5e94\u540c\u4eba\u7c7b\u4e00\u6837\uff1a\u4e3b\u89c2\u3001\u611f\u6027\u3001\u5229\u5bb3\u53d6\u820d\u3002<\/span><\/p>\n


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\u574e\u8d1d\u5c14\u8bf4\uff1a\u201c\u8bb8\u591a\u6570\u5b66\u8bfb\u7269\u7684\u57fa\u8c03\u662f\uff0c\u6570\u5b66\u5bb6\u4eec\u662f\u5728\u5bfb\u627e\u95ea\u95ea\u53d1\u5149\u7684\u5b8c\u7f8e\u771f\u7406\u3002\u5176\u5b9e\u4e0d\u662f\u8fd9\u6837\u7684\u3002\u4ed6\u4eec\u6709\u81ea\u5df1\u7684\u54c1\u5473\u548c\u89c9\u5f97\u8212\u9002\u7684\u9886\u57df\uff0c\u4ed6\u4eec\u4f1a\u56e0\u4e3a\u5ba1\u7f8e\u6216\u4e2a\u4eba\u539f\u56e0\u6452\u5f03\u81ea\u5df1\u4e0d\u559c\u6b22\u7684\u4e1c\u897f\u3002\u201d<\/span><\/p>\n


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\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u6765\u8bf4\uff0c\u5362\u91cc\u7684\u5de5\u4f5c\u5e26\u6765\u4e86\u4e00\u4e2a\u5927\u6311\u6218\u3002\u751a\u81f3\u53ef\u4ee5\u8bf4\u662f\u4e00\u79cd\u6311\u8845\uff1a\u8fd9\u91cc\u6709\u4e00\u79cd\u66f4\u597d\u7684\u7814\u7a76\u6570\u5b66\u7684\u65b9\u6cd5\u3002\u5bf9\u4e8e\u90a3\u4e9b\u804c\u4e1a\u751f\u6daf\u4e2d\u4e00\u76f4\u81f4\u529b\u4e8e\u7814\u7a76\u88ab\u5362\u91cc\u7684\u5de5\u4f5c\u8d85\u8d8a\u7684\u65b9\u6cd5\u7684\u6570\u5b66\u5bb6\u4eec\u6765\u8bf4\uff0c\u5c24\u5176\u5982\u6b64\u3002<\/span><\/p>\n


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\u5f17\u6717\u897f\u65af\u8bf4\uff1a\u201c\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\u5b58\u5728\u4e00\u79cd\u7d27\u5f20\uff0c\u4eba\u4eec\u5e76\u4e0d\u603b\u662f\u4e50\u4e8e\u770b\u5230\u4e0b\u4e00\u4ee3\u91cd\u5199\u4ed6\u4eec\u7684\u4f5c\u54c1\u3002\u8fd9\u4e00\u70b9\u5bf9\u65e0\u7a77\u8303\u7574\u8bba\u6709\u5f71\u54cd\uff0c\u4ee5\u524d\u7684\u8bb8\u591a\u5de5\u4f5c\u90fd\u88ab\u91cd\u5199\u4e86\u3002\u201d<\/span><\/p>\n


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\u9664\u6b64\u4e4b\u5916\uff0c\u5176\u4ed6\u4e00\u4e9b\u56e0\u7d20\u4e5f\u5bfc\u81f4\u5362\u91cc\u7684\u4f5c\u54c1\u5f88\u96be\u6d88\u5316\u3002\u957f\u7bc7\u5927\u4f5c\u610f\u5473\u7740\u6570\u5b66\u5bb6\u4eec\u9700\u8981\u82b1\u8d39\u5927\u91cf\u65f6\u95f4\u6765\u9605\u8bfb\u3002\u5bf9\u4e8e\u5904\u4e8e\u804c\u4e1a\u751f\u6daf\u4e2d\u671f\u7684\u5fd9\u788c\u6570\u5b66\u5bb6\u6765\u8bf4\uff0c\u8fd9\u51e0\u4e4e\u662f\u4e0d\u53ef\u80fd\u5b8c\u6210\u7684\u4efb\u52a1\uff0c\u800c\u5bf9\u4e8e\u7814\u7a76\u751f\u6765\u8bf4\uff0c\u4ed6\u4eec\u53ea\u6709\u51e0\u5e74\u65f6\u95f4\u6765\u505a\u51fa\u80fd\u8ba9\u81ea\u5df1\u627e\u5230\u5de5\u4f5c\u7684\u6210\u679c\uff0c\u56e0\u800c\u8fd9\u4e5f\u662f\u4e00\u4e2a\u9ad8\u98ce\u9669\u8981\u6c42\u3002<\/span><\/p>\n


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\u53e6\u5916\u5362\u91cc\u7684\u5de5\u4f5c\u662f\u9ad8\u5ea6\u62bd\u8c61\u7684\uff0c\u5373\u4f7f\u4e0e\u9ad8\u7b49\u6570\u5b66\u4e2d\u5176\u4ed6\u9ad8\u5ea6\u62bd\u8c61\u7684\u5185\u5bb9\u76f8\u6bd4\u4e5f\u662f\u5982\u6b64\u3002\u5c31\u53ef\u63a5\u7eb3\u7a0b\u5ea6\u800c\u8a00\uff0c\u5b83\u5e76\u4e0d\u9002\u5408\u6240\u6709\u4eba\u3002\u574e\u8d1d\u5c14\u8bf4\uff1a\u201c\u4e00\u4e9b\u4eba\u8ba4\u4e3a\u5362\u91cc\u7684\u4e66\u662f\u62bd\u8c61\u7684\u5e9f\u8bdd\uff0c\u4e00\u4e9b\u4eba\u5219\u7518\u4e4b\u5982\u9974\uff0c\u4e00\u4e9b\u4eba\u7684\u53cd\u5e94\u4ecb\u4e8e\u4e24\u8005\u4e4b\u95f4\uff0c\u4e00\u4e9b\u4eba\u5219\u662f\u5b8c\u5168\u4e0d\u7406\u89e3\u3002\u201d<\/span><\/p>\n


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\u79d1\u5b66\u754c\u4e00\u76f4\u5728\u5438\u6536\u65b0\u601d\u60f3\uff0c\u4f46\u901a\u5e38\u5f88\u7f13\u6162\uff0c\u611f\u89c9\u5c31\u50cf\u4e00\u5927\u7fa4\u4eba\u5728\u4e00\u8d77\u884c\u8fdb\u3002\u5f53\u5927\u7684\u65b0\u60f3\u6cd5\u51fa\u73b0\u65f6\uff0c\u4f1a\u5bf9\u79d1\u5b66\u754c\u7684\u77e5\u8bc6\u5438\u6536\u673a\u5236\u6784\u6210\u6311\u6218\u3002\u574e\u8d1d\u5c14\u8bf4\uff1a\u201c\u4e00\u6b21\u6027\u5bfc\u5165\u4e86\u592a\u591a\u4e1c\u897f\uff0c\u6709\u70b9\u50cf\u87d2\u86c7\u8bd5\u56fe\u541e\u4e0b\u4e00\u5934\u725b\u3002\u6709\u4e00\u5927\u56e2\u4e1c\u897f\u6b63\u5728\u901a\u8fc7\u79d1\u5b66\u754c\u3002\u201d<\/span><\/p>\n


<\/p>\n

\u5982\u679c\u4f60\u662f\u6570\u5b66\u5bb6\uff0c\u8ba4\u4e3a\u5362\u91cc\u7684\u65b9\u6cd5\u662f\u89e3\u51b3\u6570\u5b66\u95ee\u9898\u7684\u66f4\u597d\u65b9\u6cd5\uff0c\u90a3\u4e48\u524d\u8fdb\u7684\u9053\u8def\u5c06\u5f88\u5b64\u72ec\u3002\u5f88\u5c11\u6709\u4eba\u8bfb\u8fc7\u5362\u91cc\u7684\u8457\u4f5c\uff0c\u6ca1\u6709\u6559\u79d1\u4e66\u5bf9\u5b83\u8fdb\u884c\u63d0\u70bc\uff0c\u4e5f\u6ca1\u6709\u7814\u8ba8\u4f1a\u8ba9\u4f60\u660e\u786e\u65b9\u5411\u3002\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u7684\u7814\u7a76\u751f\u5f7c\u5f97\u00b7\u6d77\u6069<\/span>\uff08Peter Haine\uff09<\/span>\u82b1\u4e86\u4e00\u5e74\u65f6\u95f4\u9605\u8bfb\u5362\u91cc\u7684\u8457\u4f5c\uff0c\u4ed6\u8bf4\uff1a\u201c\u8ba9\u4f60\u80fd\u771f\u6b63\u5b66\u4f1a\u8fd9\u4e9b\u4e1c\u897f\u7684\u65b9\u5f0f\uff0c\u5c31\u662f\u5750\u4e0b\u6765\u81ea\u5df1\u52a8\u624b\u3002\u6211\u8ba4\u4e3a\u8fd9\u662f\u6700\u56f0\u96be\u7684\u3002\u4e0d\u4ec5\u4ec5\u662f\u5750\u4e0b\u6765\u81ea\u5df1\u52a8\u624b\uff0c\u800c\u662f\u901a\u8fc7\u5750\u4e0b\u6765\u8bfb800\u591a\u9875\u7684\u300a\u9ad8\u9636\u8303\u7574\u8bba\u300b\u6765\u81ea\u5df1\u505a\u3002\u201d<\/span><\/p>\n


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\u540c\u8bb8\u591a\u65b0\u53d1\u660e\u4e00\u6837\uff0c\u300a\u9ad8\u9636\u8303\u7574\u8bba\u300b\u8981\u6c42\u6570\u5b66\u5bb6\u4eec\u4e0e\u9a71\u52a8\u7406\u8bba\u7684\u673a\u5236\u8fdb\u884c\u5927\u91cf\u4e92\u52a8\u3002\u8fd9\u5c31\u50cf\u8981\u6c4216\u5c81\u7684\u5b69\u5b50\u5148\u5f97\u5b66\u4f1a\u6539\u88c5\u5f15\u64ce\u624d\u80fd\u62ff\u5230\u9a7e\u7167\u3002\u4e0e\u5362\u91cc\u5408\u4f5c\u7684\u54c8\u4f5b\u6570\u5b66\u5bb6\u4e39\u5c3c\u65af\u00b7\u76d6\u7279\u683c\u91cc<\/span>\uff08Dennis Gaitsgory\uff09<\/span>\u8bf4\uff0c\u201c\u5982\u679c\u6709\u4e00\u4e2a\u5bf9\u9a7e\u9a76\u5458\u66f4\u53cb\u597d\u7684\u7248\u672c\uff0c\u5c31\u66f4\u5bb9\u6613\u88ab\u66f4\u5e7f\u6cdb\u7684\u6570\u5b66\u89c2\u4f17\u7406\u89e3\u3002\u201d<\/span><\/p>\n


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\u968f\u7740\u4eba\u4eec\u5f00\u59cb\u9605\u8bfb\u5362\u91cc\u7684\u8457\u4f5c\uff0c\u5e76\u5728\u81ea\u5df1\u7684\u7814\u7a76\u4e2d\u4f7f\u7528\u65e0\u7a77\u8303\u7574\uff0c\u5176\u4ed6\u95ee\u9898\u4e5f\u51fa\u73b0\u4e86\u3002\u6570\u5b66\u5bb6\u4eec\u4f1a\u7528\u65e0\u7a77\u8303\u7574\u6765\u5199\u8bba\u6587\u3002\u671f\u520a\u5ba1\u7a3f\u4eba\u6536\u5230\u8fd9\u4e9b\u6587\u7ae0\u540e\u4f1a\u8bf4\uff1a\u8fd9\u662f\u4ec0\u4e48\uff1f<\/span><\/p>\n


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\u5c3d\u7ba1\u6570\u5b66\u5f88\u5f62\u5f0f\u5316\uff0c\u4f46\u6570\u5b66\u5e76\u4e0d\u662f\u53ea\u9700\u8981\u53ea\u6709\u7267\u5e08\u624d\u80fd\u9605\u8bfb\u7684\u795e\u5723\u7ecf\u6587\u3002\u8fd9\u4e2a\u9886\u57df\u65e2\u9700\u8981\u5c0f\u518c\u5b50\uff0c\u4e5f\u9700\u8981\u5927\u90e8\u5934\u7684\u4e66\u7c4d\uff0c\u9664\u4e86\u539f\u521d\u7684\u542f\u793a\uff0c\u8fd8\u9700\u8981\u89e3\u91ca\u6027\u4f5c\u54c1\u3002<\/strong>\u73b0\u5728\uff0c\u300a\u65e0\u7a77\u8303\u7574\u8bba\u300b\u4ecd\u7136\u4e3b\u8981\u4ee5\u4e66\u67b6\u4e0a\u7684\u5927\u90e8\u5934\u7684\u5f62\u5f0f\u5b58\u5728\u3002<\/span><\/p>\n


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\u745e\u6cfd\u514b\u8bf4\uff1a\u201c\u4f60\u53ef\u4ee5\u91c7\u53d6\u2018\u96c5\u5404\u5e03\u544a\u8bc9\u4e86\u4f60\u8be5\u505a\u4ec0\u4e48\uff0c\u8fd9\u5c31\u591f\u4e86\u2019\u7684\u6001\u5ea6\u3002\u4f60\u4e5f\u53ef\u4ee5\u8bf4\uff0c\u2018\u6211\u4eec\u4e0d\u77e5\u9053\u5982\u4f55\u9002\u5f53\u5730\u8868\u8ff0\u8fd9\u4e2a\u4e3b\u9898\uff0c\u4ee5\u4fbf\u4eba\u4eec\u53ef\u4ee5\u62ff\u8d77\u5b83\u5e76\u8fd0\u7528\u5b83\u3002\u2019\u201d<\/span><\/p>\n


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\u7136\u800c\uff0c\u4e00\u4e9b\u6570\u5b66\u5bb6\u51b3\u5b9a\u8fce\u63a5\u6311\u6218\uff0c\u8ba9\u66f4\u591a\u7684\u4eba\u5728\u4ed6\u4eec\u7684\u9886\u57df\u91cc\u53ef\u4ee5\u628a\u65e0\u7a77\u8303\u7574\u4f5c\u4e3a\u4e00\u79cd\u6280\u672f\u52a0\u4ee5\u5e94\u7528\u3002<\/span><\/p>\n


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4.\u5bf9\u7528\u6237\u53cb\u597d\u7684\u7406\u8bba<\/strong><\/strong><\/span><\/p>\n<\/section>\n

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\u4e3a\u4e86\u5c06\u65e0\u7a77\u8303\u7574\u8f6c\u5316\u4e3a\u53ef\u4ee5\u505a\u771f\u6b63\u6570\u5b66\u7814\u7a76\u7684\u5bf9\u8c61\uff0c\u5362\u91cc\u5fc5\u987b\u8bc1\u660e\u6709\u5173\u5b83\u4eec\u7684\u5b9a\u7406\u3002\u4e3a\u4e86\u505a\u5230\u8fd9\u4e00\u70b9\uff0c\u4ed6\u5fc5\u987b\u9009\u62e9\u4e00\u4e2a\u80cc\u666f\u6765\u5efa\u7acb\u8fd9\u4e9b\u8bc1\u660e\uff0c\u5c31\u50cf\u7814\u7a76\u51e0\u4f55\u7684\u4eba\u5fc5\u987b\u9009\u62e9\u4e00\u4e2a\u5750\u6807\u7cfb\u4e00\u6837\u3002\u6570\u5b66\u5bb6\u4eec\u79f0\u4e4b\u4e3a\u9009\u62e9\u6a21\u578b\u3002<\/span><\/p>\n


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\u5362\u91cc\u5728\u62df\u8303\u7574<\/span>\uff08quasi-categories\uff09<\/span>\u6a21\u578b\u4e2d\u53d1\u5c55\u4e86\u65e0\u7a77\u8303\u7574\u3002\u5176\u4ed6\u6570\u5b66\u5bb6\u4ee5\u524d\u66fe\u7ecf\u5728\u4e0d\u540c\u7684\u6a21\u578b\u4e2d\u53d1\u5c55\u4e86\u65e0\u7a77\u8303\u7574\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u5c1d\u8bd5\u8fdc\u6ca1\u6709\u5362\u91cc\u90a3\u4e48\u5168\u9762\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u5b83\u4eec\u66f4\u5bb9\u6613\u5904\u7406\u3002\u201c\u96c5\u5404\u5e03\u9009\u62e9\u4e86\u4e00\u4e2a\u6a21\u578b\uff0c\u5e76\u68c0\u67e5\u4e86\u5728\u8fd9\u4e2a\u6a21\u578b\u4e2d\u662f\u5426\u4e00\u5207\u90fd\u6210\u7acb\uff0c\u4f46\u8fd9\u5f80\u5f80\u4e0d\u662f\u6700\u5bb9\u6613\u7684\u6a21\u578b\uff0c\u201d\u624e\u54c8\u91cc\u7ef4\u5947\u8bf4\u3002<\/span><\/p>\n


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\u5728\u51e0\u4f55\u5b66\u4e2d\uff0c\u6570\u5b66\u5bb6\u4eec\u7cbe\u786e\u5730\u77e5\u9053\u5982\u4f55\u5728\u5750\u6807\u7cfb\u4e4b\u95f4\u5207\u6362\u3002\u4ed6\u4eec\u8fd8\u8bc1\u660e\u4e86\u5982\u679c\u5b9a\u7406\u5728\u4e00\u4e2a\u5750\u6807\u7cfb\u4e2d\u88ab\u8bc1\u660e\uff0c\u5219\u5b83\u5728\u53e6\u4e00\u4e2a\u5750\u6807\u7cfb\u4e2d\u4e5f\u6210\u7acb\u3002<\/span><\/p>\n


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\u5bf9\u4e8e\u65e0\u7a77\u8303\u7574\uff0c\u6ca1\u6709\u8fd9\u6837\u7684\u4fdd\u8bc1\u3002\u7136\u800c\uff0c\u5f53\u6570\u5b66\u5bb6\u4eec\u4f7f\u7528\u65e0\u7a77\u8303\u7574\u64b0\u5199\u8bba\u6587\u65f6\uff0c\u4ed6\u4eec\u5f80\u5f80\u8f7b\u7387\u5730\u5728\u6a21\u578b\u4e4b\u95f4\u5207\u6362\uff0c\u5047\u8bbe\uff08\u4f46\u4e0d\u8bc1\u660e\uff09\u4ed6\u4eec\u7684\u7ed3\u679c\u53ef\u4ee5\u4fdd\u6301\u6210\u7acb\u3002\u6d77\u6069\u8bf4\uff1a\u201c\u4eba\u4eec\u4e0d\u4f1a\u8be6\u7ec6\u8bf4\u660e\u4ed6\u4eec\u5728\u505a\u4ec0\u4e48\uff0c\u4ed6\u4eec\u4f1a\u5728\u8fd9\u4e9b\u4e0d\u540c\u7684\u6a21\u578b\u4e4b\u95f4\u5207\u6362\uff0c\u7136\u540e\u8bf4\uff0c\u2018\u54e6\uff0c\u90fd\u662f\u4e00\u6837\u7684\u2019\u3002\u4f46\u8fd9\u4e0d\u662f\u8bc1\u660e\u3002\u201d<\/span><\/p>\n


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\u5728\u8fc7\u53bb\u516d\u5e74\u91cc\uff0c\u4e24\u4f4d\u6570\u5b66\u5bb6\u4e00\u76f4\u5728\u52aa\u529b\u505a\u51fa\u8fd9\u6837\u7684\u4fdd\u8bc1\u3002\u91cc\u5c14\u548c\u6765\u81ea\u6fb3\u5927\u5229\u4e9a\u9ea6\u89c9\u7406\u5927\u5b66<\/span>\uff08Macquarie University\uff09\u7684<\/span>\u591a\u7c73\u5c3c\u514b\u00b7\u7ef4\u91cc\u8482<\/span>\uff08Dominic Verity\uff09<\/span>\u4e00\u76f4\u5728\u7814\u7a76\u4e00\u79cd\u63cf\u8ff0\u65e0\u7a77\u8303\u7574\u7684\u65b9\u6cd5\uff0c\u8fd9\u79cd\u65b9\u6cd5\u8d85\u8d8a\u4e86\u4ee5\u524d\u9650\u5b9a\u4e8e\u6a21\u578b\u7684\u6846\u67b6\u6240\u9020\u6210\u7684\u56f0\u96be\u3002\u4ed6\u4eec\u7684\u5de5\u4f5c\u5efa\u7acb\u5728\u5df4\u7ef4\u514b\u7b49\u4eba\u4e4b\u524d\u7684\u5de5\u4f5c\u7684\u57fa\u7840\u4e0a\uff0c\u5df2\u7ecf\u8bc1\u660e\u4e86\u9ad8\u9636\u8303\u7574\u8bba\u4e2d\u7684\u8bb8\u591a\u5b9a\u7406\u90fd\u662f\u6210\u7acb\u7684\uff0c\u65e0\u8bba\u4f60\u5c06\u5b83\u4eec\u5e94\u7528\u4e8e\u54ea\u4e2a\u6a21\u578b\u4e2d\u3002<\/strong>\u4ed6\u4eec\u7528\u4e00\u79cd\u6070\u5f53\u7684\u65b9\u5f0f\u8bc1\u660e\u4e86\u8fd9\u79cd\u517c\u5bb9\u6027\uff0c\u91cc\u5c14\u89e3\u91ca\u8bf4\uff1a\u201c\u6211\u4eec\u6b63\u5728\u7814\u7a76\u7684\u65e0\u7a77\u8303\u7574\uff0c\u5176\u5bf9\u8c61\u672c\u8eab\u5c31\u662f\u65e0\u7a77\u8303\u7574\u3002\u8303\u7574\u8bba\u5728\u8fd9\u91cc\u5c31\u50cf\u5728\u541e\u98df\u81ea\u5df1\u3002\u201d<\/span><\/p>\n


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\u7ea6\u7ff0\u00b7\u970d\u666e\u91d1\u65af\u5927\u5b66\u6570\u5b66\u5bb6\u827e\u7c73\u4e3d\u00b7\u91cc\u5c14\uff08Emily Riehl\uff09\u6b63\u5728\u5e2e\u52a9\u5f15\u5bfc\u9ad8\u9636\u8303\u7574\u7406\u8bba\u7684\u53d1\u5c55\u3002| \u56fe\u7247\u6765\u6e90\uff1aWill Kirk \/\u7ea6\u7ff0\u00b7\u970d\u666e\u91d1\u65af\u5927\u5b66<\/span><\/p>\n


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\u91cc\u5c14\u548c\u7ef4\u91cc\u8482\u8fd8\u5e0c\u671b\u4ee5\u53e6\u4e00\u79cd\u65b9\u5f0f\u63a8\u52a8\u65e0\u7a77\u8303\u7574\u8bba\u7684\u53d1\u5c55\u3002\u4ed6\u4eec\u9610\u660e\u4e86\u65e0\u7a77\u8303\u7574\u8bba\u65e0\u8bba\u5728\u54ea\u4e2a\u6a21\u578b\u4e2d\u90fd\u6210\u7acb\u7684\u90a3\u4e9b\u7279\u6027\u3002\u8fd9\u79cd\u201c\u4e0e\u6a21\u578b\u65e0\u5173\u201d\u7684\u8868\u793a\u5177\u6709\u5373\u63d2\u5373\u7528\u7684\u7279\u6027\uff0c\u4ed6\u4eec\u5e0c\u671b\uff0c\u8fd9\u79cd\u7279\u6027\u80fd\u591f\u8ba9\u90a3\u4e9b\u6700\u521d\u53ea\u80fd\u901a\u8fc7\u300a\u9ad8\u9636\u8303\u7574\u8bba\u300b\u8fdb\u5165\u8fd9\u4e2a\u9886\u57df\u56e0\u800c\u79bb\u5f00\u7684\u6570\u5b66\u5bb6\u80fd\u56de\u5230\u8fd9\u4e2a\u9886\u57df\u3002<\/span><\/p>\n


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\u201c\u8981\u8fdb\u5165\u8fd9\u4e2a\u4e16\u754c\uff0c\u4f60\u5fc5\u987b\u7a7f\u8fc7\u4e00\u6761\u62a4\u57ce\u6cb3\uff0c\u800c\u4ed6\u4eec\u6b63\u5728\u653e\u4e0b\u540a\u6865\u3002\u201d\u970d\u666e\u91d1\u65af\u8bf4\u9053\u3002<\/span><\/p>\n


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\u91cc\u5c14\u548c\u7ef4\u91cc\u8482\u5e0c\u671b\u80fd\u5728\u660e\u5e74\u5b8c\u6210\u4ed6\u4eec\u7684\u5de5\u4f5c\u3002\u4e0e\u6b64\u540c\u65f6\uff0c\u5362\u91cc\u6700\u8fd1\u5f00\u59cb\u4e86\u4e00\u4e2a\u540d\u4e3a\u5ca9\u8c5a\u9f20<\/span>\uff08Kerodon\uff09<\/span>\u7684\u9879\u76ee\uff0c\u4ed6\u6253\u7b97\u628a\u8fd9\u4e2a\u9879\u76ee\u4f5c\u4e3a\u7ef4\u57fa\u767e\u79d1\u5f0f\u7684\u9ad8\u9636\u8303\u7574\u8bba\u6559\u79d1\u4e66\u3002\u5728\u300a\u9ad8\u9636\u8303\u7574\u8bba\u300b\u4f7f\u7b49\u4ef7\u7684\u6570\u5b66\u5f62\u5f0f\u531613\u5e74\u4e4b\u540e\uff0c\u8fd9\u4e9b\u5021\u8bae\u662f\u63d0\u70bc\u548c\u63a8\u5e7f\u8fd9\u4e2a\u601d\u60f3\u7684\u65b0\u5c1d\u8bd5\u2014\u2014\u4f7f\u7b49\u4ef7\u7684\u6570\u5b66\u66f4\u52a0\u666e\u53ca\u3002<\/span><\/p>\n


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\u5ca9\u8c5a\u9f20\uff08Kerodon\uff09\u9879\u76ee\u3002| \u56fe\u7247\u6765\u6e90\uff1akerodon.net<\/span><\/p>\n


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\u4e54\u4e9a\u5c14\u8bf4\uff1a\u201c\u5929\u624d\u5728\u6570\u5b66\u53d1\u5c55\u4e2d\u626e\u6f14\u4e86\u91cd\u8981\u7684\u89d2\u8272\uff0c\u4f46\u5b9e\u9645\u4e0a\u77e5\u8bc6\u672c\u8eab\u662f\u5b66\u672f\u754c\u6d3b\u52a8\u7684\u7ed3\u679c\u3002\u77e5\u8bc6\u7684\u771f\u6b63\u76ee\u7684\u662f\u6210\u4e3a\u793e\u533a\u7684\u77e5\u8bc6\uff0c\u800c\u4e0d\u662f\u4e00\u4e24\u4e2a\u4eba\u7684\u77e5\u8bc6\u3002\u201d<\/span><\/p>\n


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\u6570\u5b66\u548c\u7269\u7406\u878d\u5408\u7684\u9ec4\u91d1\u65f6\u4ee3<\/span><\/strong><\/span><\/p>\n


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\u64b0\u6587 | \u6587\u5c0f\u521a \uff08\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u7ec8\u8eab\u6559\u6388\u3001\u683c\u6797\u8bb2\u5e2d\u6559\u6388\uff09<\/span><\/p>\n


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\u7269\u7406\u5b66\u7684\u76ee\u7684\u662f\u51c6\u786e\u5730\u7406\u89e3\u548c\u63cf\u8ff0\u5404\u79cd\u5404\u6837\u7684\u81ea\u7136\u73b0\u8c61\u3002\u4f46\u6211\u4eec\u7684\u7269\u7406\u4e16\u754c\u662f\u5982\u6b64\u4e30\u5bcc\u591a\u5f69\uff0c\u4f7f\u6211\u4eec\u65e0\u6cd5\u7528\u65e5\u5e38\u751f\u6d3b\u6240\u53d1\u5c55\u51fa\u6765\u7684\u8bed\u8a00\u6765\u51c6\u786e\u63cf\u5199\u8fd9\u4e9b\u81ea\u7136\u73b0\u8c61\u3002\u7279\u522b\u662f\u5f53\u6211\u4eec\u53d1\u73b0\u4e00\u7c7b\u5168\u65b0\u7684\u81ea\u7136\u73b0\u8c61\u65f6\uff0c\u7269\u7406\u5b66\u5bb6\u5e38\u5e38\u53d1\u73b0\u4ed6\u4eec\u9700\u8981\u5f15\u5165\u65b0\u6570\u5b66\u3001\u53d1\u660e\u65b0\u8bed\u8a00\u6765\u63cf\u5199\u8fd9\u4e9b\u81ea\u7136\u73b0\u8c61\u3002<\/span><\/p>\n


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\u50cf\u725b\u987f\u9700\u8981\u53d1\u660e\u5fae\u79ef\u5206\u6765\u63cf\u5199\u4ed6\u7684\u529b\u5b66\u7406\u8bba\u6240\u63cf\u5199\u7684\u66f2\u7ebf\u8fd0\u52a8\u3002\u4e3a\u4e86\u63cf\u5199\u7535\u78c1\u73b0\u8c61\uff0c\u6211\u4eec\u9700\u8981\u7528\u5230\u6570\u5b66\u4e2d\u7684\u7ea4\u7ef4\u4e1b\u7406\u8bba\uff0c\u800c\u4e3a\u4e86\u63cf\u5199\u5f15\u529b\u73b0\u8c61\uff0c\u6211\u4eec\u9700\u8981\u9ece\u66fc\u51e0\u4f55\u7406\u8bba\u3002\u5f53\u6211\u4eec\u53d1\u73b0\u5fae\u89c2\u4e16\u754c\u7684\u91cf\u5b50\u73b0\u8c61\u540e\uff0c\u6211\u4eec\u610f\u8bc6\u5230\u63cf\u5199\u6211\u4eec\u4e16\u754c\u7684\u6570\u5b66\u7406\u8bba\u5e76\u4e0d\u662f\u5fae\u79ef\u5206\u3001\u7ea4\u7ef4\u4e1b\u548c\u9ece\u66fc\u51e0\u4f55\uff0c\u800c\u662f\u5e26\u5f20\u91cf\u4e58\u6cd5\u7684\u7ebf\u6027\u4ee3\u6570\u3002<\/span><\/p>\n


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\u5728\u51dd\u805a\u6001\u7269\u7406\u548c\u6750\u6599\u79d1\u5b66\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u7406\u89e3\u548c\u63cf\u5199\u5343\u5343\u4e07\u4e07\u3001\u5404\u79cd\u5404\u6837\u7684\u7269\u8d28\u6001\u3002\u6717\u9053\u4ee5\u4ed6\u6df1\u523b\u7684\u6d1e\u5bdf\u6307\u51fa\uff0c\u5404\u79cd\u5404\u6837\u7684\u7269\u8d28\u6001\u5176\u5b9e\u6765\u6e90\u4e8e\u5b83\u4eec\u5185\u90e8\u5404\u79cd\u5404\u6837\u4e0d\u540c\u7684\u5bf9\u79f0\u6027\u7834\u7f3a\u3002\u4e8e\u662f\uff0c\u63cf\u5199\u5bf9\u79f0\u6027\u7684\u7fa4\u8bba\u5c31\u6210\u4e3a\u6211\u4eec\u63cf\u5199\u5404\u79cd\u5404\u6837\u7269\u8d28\u6001\u7684\u6570\u5b66\u8bed\u8a00\u3002<\/span><\/p>\n


<\/p>\n

\u53ef\u662f\u8fc7\u53bb30\u5e74\u6765\u51dd\u805a\u6001\u7269\u7406\u7684\u8fdb\u5c55\u63ed\u793a\u4e86\u4e00\u5927\u7c7b\u5168\u65b0\u7269\u8d28\u6001\u7684\u5b58\u5728\u3002\u8fd9\u4e00\u7c7b\u7269\u8d28\u6001\u4e0d\u662f\u8d77\u6e90\u4e8e\u5bf9\u79f0\u6027\uff0c\u800c\u662f\u8d77\u6e90\u4e8e\u6750\u6599\u4e2d\u7684\u591a\u4f53\u91cf\u5b50\u7ea0\u7f20\u3002\u591a\u4f53\u91cf\u5b50\u7ea0\u7f20<\/span>\uff08\u4e5f\u5c31\u662f\u62d3\u6251\u5e8f\uff09<\/span>\u662f\u4e00\u79cd\u5168\u65b0\u7684\u81ea\u7136\u73b0\u8c61\u3002\u6211\u4eec\u5230\u5e95\u5e94\u8be5\u7528\u4ec0\u4e48\u6837\u7684\u6570\u5b66\u8bed\u8a00\u624d\u80fd\u63cf\u5199\u8fd9\u79cd\u5168\u65b0\u7684\u81ea\u7136\u73b0\u8c61\u5462\uff1f<\/span><\/p>\n


<\/p>\n

\u4e3a\u4e86\u7406\u89e3\u548c\u63cf\u8ff0\u591a\u4f53\u91cf\u5b50\u7ea0\u7f20\u7684\u5185\u90e8\u7ed3\u6784\uff0c\u4e5f\u5c31\u662f\u62d3\u6251\u5e8f\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u8fd9\u4e00\u7ed3\u6784\u6240\u5141\u8bb8\u7684\u5404\u79cd\u5404\u6837\u7684\u70b9\u7f3a\u9677\uff0c\u5e76\u901a\u8fc7\u8fd9\u4e9b\u7f3a\u9677\u7684\u6027\u8d28\u6765\u7406\u89e3\u548c\u63cf\u8ff0\u8fd9\u4e00\u7ed3\u6784\u3002\u4f46\u4e00\u4e2a\u62d3\u6251\u5e8f\u53ef\u4ee5\u6709\u65e0\u7a77\u591a\u4e2a\u4e0d\u540c\u7684\u7f3a\u9677\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u65e0\u7a77\u5927\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u91cd\u65b0\u5b9a\u4e49\u4ec0\u4e48\u53eb\u505a\u201c\u76f8\u7b49\u201d\uff1a\u5f53\u4e24\u4e2a\u7f3a\u9677\u53ef\u4ee5\u901a\u8fc7\u5c40\u90e8\u5f62\u53d8\u800c\u76f8\u4e92\u8f6c\u6362\u65f6\uff0c\u6211\u4eec\u79f0\u5b83\u4eec\u662f\u7b49\u4ef7\u7684\uff0c\u6216\u8005\u662f\u201c\u76f8\u7b49\u201d\u7684\u3002\u6211\u4eec\u53d1\u73b0\u62d3\u6251\u5e8f\u4e2d\u7684\u7f3a\u9677\u53ea\u6709\u6709\u9650\u591a\u4e2a\u7b49\u4ef7\u7c7b\u3002\u8fd9\u4e9b\u4e0d\u540c\u7c7b\u578b\u7f3a\u9677\u7684\u7b49\u4ef7\u5173\u7cfb\u53ef\u4ee5\u662f\u975e\u5e38\u590d\u6742\u7684\uff0c\u56e0\u4e3a\u5b83\u4eec\u5305\u62ec\u7f3a\u9677\u4e4b\u95f4\u7684\u878d\u5408\u3001\u7f16\u7ec7\u7b49\u7b49\u5c40\u90e8\u64cd\u4f5c\u3002\u63cf\u5199\u8fd9\u4e9b\u7f3a\u9677\u7b49\u4ef7\u7684\u7c7b\u7684\u6570\u5b66\u7406\u8bba\u6b63\u662f\u672c\u6587\u6240\u63cf\u8ff0\u7684\u8303\u7574\u8bba\u3002\u8303\u7574\u8bba\u8fd9\u4e00\u6781\u7aef\u62bd\u8c61\u7684\u6570\u5b66\u5c31\u8fd9\u6837\u8d70\u8fdb\u4e86\u51dd\u805a\u6001\u7269\u7406\u3002<\/span><\/p>\n


<\/p>\n

\u5176\u5b9e\u62d3\u6251\u5e8f\u4e2d\u7684\u7f3a\u9677\u4e0d\u4ec5\u53ef\u4ee5\u662f\u70b9\u72b6\u7684\uff0c\u8fd8\u53ef\u4ee5\u662f\u7ebf\u72b6\u7684\u3001\u9762\u72b6\u7684\u7b49\u7b49\u3002\u63cf\u5199\u8fd9\u4e9b\u66f4\u590d\u6742\u7f3a\u9677\u7684\u7b49\u4ef7\u7c7b\u7684\u6570\u5b66\u8bed\u8a00\u5c31\u662f\u672c\u6587\u6240\u4ecb\u7ecd\u7684\u9ad8\u9636\u8303\u7574\u5b66\uff0c\u6216\u65e0\u7a77\u8303\u7574\u5b66\u3002\u8fd9\u4e9b\u62bd\u8c61\u6570\u5b66\u7406\u8bba\u662f\u63cf\u5199\u591a\u4f53\u91cf\u5b50\u7ea0\u7f20\u8fd9\u4e00\u5168\u65b0\u7269\u7406\u73b0\u8c61\u7684\u81ea\u7136\u8bed\u8a00\u3002\u65b0\u7684\u6570\u5b66\u8fdb\u5165\u7269\u7406\u610f\u5473\u7740\u7269\u7406\u7684\u65b0\u9769\u547d\u3002\u73b0\u5728\u6b63\u662f\u6570\u5b66\u548c\u7269\u7406\u9ad8\u901f\u53d1\u5c55\u7684\u9ec4\u91d1\u65f6\u671f\u3002<\/span><\/p>\n

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<\/p>\n<\/section>\n<\/section>\n

\u64b0\u6587 | \u5b54\u826f\uff08\u6df1\u5733\u91cf\u5b50\u79d1\u5b66\u4e0e\u5de5\u7a0b\u7814\u7a76\u9662\uff0c\u5357\u65b9\u79d1\u6280\u5927\u5b66\uff09<\/span><\/p>\n


<\/p>\n

\u6beb\u65e0\u7591\u95ee Jacob Lurie \u7684\u5de5\u4f5c\u503c\u5f97\u5355\u72ec\u64b0\u6587\u6765\u8ba8\u8bba\uff0c\u4f46\u662f\u501f\u8fd9\u7bc7\u6587\u7ae0\u7684\u4e1c\u98ce\uff0c\u52a0\u4e00\u4e9b\u7b80\u6d01\u7684\u8bc4\u8bba\u4e5f\u53ef\u80fd\u662f\u6709\u76ca\u7684\u3002<\/span><\/p>\n


<\/p>\n

Jacob Lurie\u7684\u4e24\u90e8\u957f\u7bc7\u5de8\u8457 Higher Topos Theory \u548c Higher Algebra \u662f\u8fd1\u8fc7\u53bb20\u5e74\u6570\u5b66\u91cc\u9762\u6700\u6fc0\u52a8\u4eba\u5fc3\u7684\u8fdb\u5c55\u4e4b\u4e00\u3002\u6211\u4eec\u77e5\u9053\u5fae\u79ef\u5206\u548c\u7ebf\u6027\u4ee3\u6570\u662f\u73b0\u4ee3\u7269\u7406\u548c\u5176\u4ed6\u79d1\u5b66\u7684\u57fa\u7840\u8bed\u8a00\u3002\u7c97\u7565\u5730\u8bb2\uff0cHigher Topos Theory \u53ef\u4ee5\u770b\u4f5c\u662f\u4e00\u79cd\u65b0\u7684\u5fae\u79ef\u5206\uff0c\u800c Higher Algebra \u662f\u4e00\u79cd\u65b0\u7684\u7ebf\u6027\u4ee3\u6570\u3002\u5b83\u4eec\u4e0d\u4ec5\u5728\u4e00\u4e2a\u5f88\u9ad8\u7684\u89c6\u89d2\u4e0a\u7edf\u4e00\u4e86\u8fc7\u53bb\u7684\u5f88\u591a\u6570\u5b66\uff0c\u800c\u4e14\u8fd8\u63d0\u4f9b\u4e86\u4e00\u5f20\u5b8f\u4f1f\u7684\u6570\u5b66\u65b0\u84dd\u56fe\u3002\u800c\u8fc7\u53bb\u5df2\u77e5\u7684\u6570\u5b66\u4f3c\u4e4e\u53ea\u662f\u8fd9\u5f20\u65b0\u84dd\u56fe\u7684\u4e00\u89d2\uff0c\u53ef\u4ee5\u6beb\u4e0d\u5938\u5f20\u5730\u8bf4\uff0c\u6570\u5b66\u624d\u521a\u521a\u5f00\u59cb\u3002 <\/span><\/p>\n


<\/p>\n

\u6709\u8da3\u7684\u662f\uff0cJacob Lurie \u7684\u957f\u6587\u5e76\u6ca1\u6709\u88ab\u6570\u5b66\u754c\u5e7f\u6cdb\u5730\u63a5\u53d7\uff0c\u9664\u975e\u53ef\u4ee5\u7528\u65b0\u8bed\u8a00\u3001\u65b0\u5de5\u5177\u6765\u5f81\u670d\u4f20\u7edf\u6570\u5b66\u5bb6\u90fd\u653b\u514b\u4e0d\u4e86\u7684\u8001\u95ee\u9898 \u3002\u4e0d\u8fc7\u5728\u6570\u5b66\u7269\u7406\u5b66\u5bb6\u7684\u5708\u5b50\u91cc\uff0cJacob Lurie \u7684\u8d21\u732e\u4e0d\u4f46\u88ab\u5e7f\u6cdb\u800c\u5feb\u901f\u5730\u63a5\u53d7\uff0c\u800c\u4e14\u5df2\u7ecf\u542f\u53d1\u4e86\u5927\u91cf\u540e\u7eed\u548c\u5e73\u884c\u7684\u5de5\u4f5c\u3002\u5f53\u7136\u8fd9\u5e76\u4e0d\u5947\u602a\uff0c\u56e0\u4e3a Lurie \u7684\u5de5\u4f5c\u672c\u6765\u5c31\u53d7\u5230\u4e86\u8fc7\u53bb30\u5e74\u7531\u5f26\u8bba\u548c\u91cf\u5b50\u573a\u8bba\u5f15\u53d1\u7684\u6570\u5b66\u7269\u7406\u65b0\u6f6e\u6d41\u7684\u6df1\u523b\u5f71\u54cd\uff0c\u7279\u522b\u662f\u62d3\u6251\u573a\u8bba\u3001mirror symmetry\u3001derived algebraic geometry\u3001En \u4ee3\u6570\u3001chiral homology \u7b49\u7b49\u3002Lurie \u91cc\u7a0b\u7891\u5f0f\u7684\u5de5\u4f5c\u4e00\u4e0b\u5b50\u5728\u4e00\u4e2a\u5f88\u9ad8\u7684\u9ad8\u5ea6\u4e0a\uff0c\u628a\u4ee5\u524d\u5f88\u591a\u96f6\u788e\u7684\u601d\u60f3\u7edf\u4e00\u5728\u4e00\u8d77\uff0c\u5e76\u542f\u53d1\u4e86\u5f88\u591a\u5168\u65b0\u7684\u95ee\u9898\u3002 <\/span><\/p>\n


<\/p>\n

\u6211\u8fd8\u60f3\u5f3a\u8c03\u7684\u662f\uff0c\u5373\u4f7f\u5728 Lurie \u7684\u5de5\u4f5c\u4e4b\u540e\uff0c\u91cf\u5b50\u573a\u8bba\u5e26\u6765\u7684\u7269\u7406\u76f4\u89c9\u4ecd\u7136\u662f\u6fc0\u53d1\u60f3\u8c61\u529b\u7684\u6e90\u6cc9\u3002\u5f88\u591a Lurie \u6ca1\u6709\u95ee\u51fa\u6765\u7684\u91cd\u8981\u95ee\u9898\u4ecd\u7136\u88ab\u4e0d\u65ad\u5730\u6316\u6398\u51fa\u6765\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c Lurie \u51e0\u5343\u9875\u7684\u6d69\u701a\u5de5\u7a0b\u4ecd\u7136\u4e0d\u8db3\u4ee5<\/span>\uff08\u54ea\u6015\u662f\u7c97\u7565\u5730\uff09<\/span>\u63cf\u7ed8\u6570\u5b66\u65b0\u84dd\u56fe\u7684\u5168\u8c8c\u3002\u5927\u81ea\u7136\u7ed9\u6211\u4eec\u7684\u542f\u8fea\u662f\u8d85\u8d8a\u60f3\u8c61\u529b\u7684\u3002<\/span><\/p>\n


<\/p>\n

\u4e0d\u8bba\u5982\u4f55\uff0c\u6211\u4eec\u90fd\u4f1a\u540c\u610f\uff0c\u8fd9\u4e2a\u65f6\u4ee3\u662f\u6570\u5b66\u548c\u7269\u7406\u878d\u5408\u7684\u9ec4\u91d1\u65f6\u4ee3\u3002<\/span><\/p>\n


<\/p>\n

\u5bf9\u4e8e\u6587\u4e2d\u63d0\u5230\u7684\u65e0\u7a77\u8303\u7574\u4ee5\u53ca Lurie \u5de5\u4f5c\u7684\u610f\u4e49\uff0c\u54e5\u5ef7\u6839\u5927\u5b66\u6570\u5b66\u6559\u6388\u6731\u6668\u7545\u7ed9\u51fa\u4e86\u5982\u4e0b\u6ce8\u91ca\uff1a<\/span><\/p>\n


<\/section>\n

\u6587\u4e2d\u7684\u65e0\u7a77\u8303\u7574\uff0c\u4e5f\u79f0\u4e3a\uff08\u221e, 1\uff09\u8303\u7574\uff0c\u56e0\u4e3a\u5b83\u7684\u9ad8\u4e8e\u7b49\u4e8e 2 \u7684 morphisms \u90fd\u662f\u53ef\u9006\u7684\u3002\u5728\u6b64\u4e4b\u524d\uff0c\u5176\u4ed6\u6570\u5b66\u5bb6\u4e5f\u66fe\u6709\u8fc7\u4e0d\u540c\u7684\u6a21\u578b\u3002\u4e0d\u540c\u7684\u5f88\u591a\u6a21\u578b\u7684\u786e\u662f\u7b49\u4ef7\u7684\uff0c\u5b83\u4eec\u4e4b\u95f4\u7684\u4e92\u76f8\u6bd4\u8f83\u4e5f\u6709\u4e00\u4e9b\u65e9\u671f\u7684\u5de5\u4f5c\uff08\u4f8b\u5982 Julia E. Bergner \u7684\u4e00\u7bc7\u7efc\u8ff0\uff1ahttps:\/\/arxiv.org\/abs\/math\/0610239 \uff09\u3002<\/span><\/p>\n


<\/section>\n

\u4f46 Lurie \u5de5\u4f5c\u7684\u610f\u4e49\u4e0d\u4ec5\u4ec5\u662f\u7ed9\u4e86\u65e0\u7a77\u8303\u7574\u4e00\u4e2a\u66f4\u5b8c\u6574\u7684\u5b9a\u4e49\u4f53\u7cfb\uff0c\u800c\u662f\u4e00\u79cd\u5c06\u65e0\u7a77\u8303\u7574\u7684\u601d\u60f3\u4f5c\u4e3a\u57fa\u7840\uff0c\u878d\u5165\u5f53\u4ee3\u6570\u5b66\u3001\u62d3\u6251\u3001\u51e0\u4f55\uff08\u4ee3\u6570\u51e0\u4f55\uff0c\u8fd9\u65b9\u9762\u4e5f\u6709\u5f88\u591a\u6765\u81ea\u6b27\u6d32\u7684 Toen \u56e2\u961f\u7684\u5de5\u4f5c\uff09\uff0c\u4ee5\u53ca\u4ee3\u6570\uff08 Operad \u7406\u8bba\uff09\u4e4b\u4e2d\u7684\u5c1d\u8bd5\u3002\u4ee5\u5fd7\u4e8e\u5168\u9762\u5730\u7ed9\u6570\u5b66\u4e00\u4e2a\u65b0\u7684\uff0c\u6216\u8005\u8bf4\u66f4\u5168\u9762\u7684\u57fa\u7840\u4f53\u7cfb\u3002<\/span><\/p>\n


<\/section>\n
\u672c\u6587\u7ffb\u8bd1\u81eaQuanta Magazine\uff0c\u539f\u6587\u6807\u9898\u4e3a\u201cWith Category Theory, Mathematics Escapes From Equality\u201d\u3002<\/span><\/section>\n

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\u8303\u7574\u8bba\u7cbe\u54c1\u5165\u95e8\u7cfb\u5217\u8bfe\u7a0b\uff0f\u6bcf\u5468\u66f4\u65b0<\/strong><\/p>\n

\u6301\u7eed\u62a5\u540d\u4e2d<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


<\/p>\n

\u8bfe\u7a0b\u5927\u7eb2\uff08\u7b2c\u4e00\u5b63\uff09\uff1a<\/span><\/strong><\/span><\/p>\n

    \n
  1. \n

    \u7ebf\u6027\u4ee3\u6570\u2014\u2014\u8303\u7574\u7684\u89c6\u89d2<\/p>\n<\/li>\n

  2. \n

    \u96c6\u5408\u8303\u7574\u548c\u7b49\u4ef7\u5173\u7cfb<\/p>\n<\/li>\n

  3. \n

    \u504f\u5e8f\u96c6\u8303\u7574<\/p>\n<\/li>\n

  4. \n

    Abel\u7fa4\u8303\u7574<\/p>\n<\/li>\n

  5. \n

    \u7ebf\u6027\u7a7a\u95f4\u7684\u8303\u7574\u5316\u6784\u9020<\/p>\n<\/li>\n

  6. \n

    Hom\u51fd\u5b50<\/p>\n<\/li>\n

  7. \n

    \u7ebf\u6027\u7a7a\u95f4\u7684\u5bf9\u5076\u6027<\/p>\n<\/li>\n

  8. \n

    \u6b63\u5411\u6781\u9650\u4e0e\u9006\u5411\u6781\u9650<\/p>\n<\/li>\n

  9. \n

    \u6b63\u5408<\/p>\n<\/li>\n

  10. \n

    \u4ece\u96c6\u5408\u5230\u62d3\u6251\u7a7a\u95f4<\/p>\n<\/li>\n

  11. \n

    \u81ea\u7531\u51fd\u5b50<\/p>\n<\/li>\n

  12. \n
    \u4ece\u51e0\u4f55\u5230\u4ee3\u6570\u2014\u2014\u540c\u8c03\u7fa4\u7684\u6784\u9020<\/section>\n<\/li>\n<\/ol>\n

    <\/section>\n

    \u8bfe\u7a0b\u76ee\u7684<\/span><\/strong><\/span><\/p>\n