{"id":30524,"date":"2021-12-01T22:27:20","date_gmt":"2021-12-01T14:27:20","guid":{"rendered":"https:\/\/swarma.org\/?p=30524"},"modified":"2021-12-01T22:27:20","modified_gmt":"2021-12-01T14:27:20","slug":"1-1-2-%ef%bc%9f-%e8%8c%83%e7%95%b4%e5%ad%a6%e7%9a%84%e5%8f%98%e9%9d%a9","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=30524","title":{"rendered":"1 + 1 = 2 \uff1f| \u8303\u7574\u5b66\u7684\u53d8\u9769"},"content":{"rendered":"
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\u5728\u8fd8\u539f\u8bba\u548c\u6f14\u2f63\u8bba\u89d2\u9010\u7684\u4eca\u5929\uff0c\u8303\u7574\u5b66\u6b63\u53d8\u5f97\u8d8a\u6765\u8d8a\u91cd\u8981\u3002\u53ef\u4ee5\u8bf4\uff0c\u5fae\u79ef\u5206\u5bf9\u8fd8\u539f\u8bba\u7684\u7269\u7406\u6709\u591a\u91cd\u8981\uff0c\u8303\u7574\u5b66\u5bf9\u6f14\u2f63\u8bba\u7684\u7269\u7406\u5c31\u6709\u591a\u91cd\u8981\u3002\u8303\u7574\u5b66\u5e26\u6765\u4e86\u8d85\u4e4e\u6570\u5b66\u5bb6\u60f3\u8c61\u7684\u6ce2\u6f9c\u58ee\u9614\uff0c\u5b83\u5728\u6982\u5ff5\u4e0a\u7edf\u2f00\u4e86\u5206\u6790\u548c\u4ee3\u6570\uff0c\u7edf\u2f00\u4e86\u79bb\u6563\u548c\u8fde\u7eed\u3002\u4eca\u5929\u6211\u4eec\u5c31\u5e26\u7740\u66fe\u7ecf\u5bf9 1+1=2 \u62b1\u6709\u7684\u597d\u5947\u2f3c\uff0c\u8fdb\u5165\u8303\u7574\u5b66\u7684\u5947\u5999\u4e16\u754c\u5427\uff01<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u8fd9\u7bc7\u2f42\u7ae0\u771f\u7684\u662f\u8bb2\u4f60\u5728\u5e7c\u2f09\u56ed\u5b66\u76841+1=2\uff0c\u4e0d\u9700\u8981\u4efb\u4f55\u6570\u5b66\u80cc\u666f\u4e5f\u53ef\u4ee5\u8bfb\uff0c\u53ea\u9700\u8981\u4f60\u6709\u5bf91+1=2\u7684\u597d\u5947\u2f3c\u3002\u4f46\u662f\u6211\u4eec\u7684\u52a8\u673a\u5374\u662f\u8981\u4ecb\u7ecd\u6570\u5b66\u2fa5\u2faf\u7684\u8303\u7574\u5b66\u7684\u57fa\u672c\u7cbe\u795e\uff0c\u6240\u4ee5\u6709\u5fc5\u8981\u5148\u7b80\u5355\u63d0\u2f00\u4e0b\u8303\u7574\u5b66\uff0c\u4e0d\u5173\u2f3c\u7684\u8bfb\u8005\u53ef\u4ee5\u76f4\u63a5\u8df3\u8fc7\u5f15\u2f94\u3002
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\u8303\u7574\u5b66\u8d77\u6e90\u4e8e\u4ee3\u6570\u62d3\u6251\uff0c\u7531Samuel Eilenberg<\/span>\uff081913-1998\uff09<\/span>\u548cSaunders Mac Lane<\/span>\uff081909-2005\uff09<\/span>\u4e8e\u4e0a\u4e2a\u4e16\u7eaa40\u5e74\u4ee3\u63d0\u51fa\u3002\u2f83\u4ece\u516d\u2f17\u5e74\u4ee3Alexander Grothendieck <\/span>\uff081928-2014\uff09<\/span>\u2f64\u8303\u7574\u5b66\u7684\u8bed\u2f94\u91cd\u5efa\u4e86\u4ee3\u6570\u2f0f\u4f55\u57fa\u7840\u4ee5\u6765\uff0c\u6570\u5b66\u4e2d\u5c31\u51fa\u73b0\u4e86\u2f64\u8303\u7574\u5b66\u66ff\u4ee3\u96c6\u5408\u8bba\u4f5c\u4e3a\u6570\u5b66\u7684\u65b0\u57fa\u7840\u7684\u6f6e\u6d41\u3002\u8fd9\u4e2a\u6f6e\u6d41\u4e0d\u4f46\u5728\u6570\u5b66\u2fa5\u2faf\u6108\u6f14\u6108\u70c8\uff0c\u8fd8\u572890\u5e74\u4ee3\u88ab\u6ce8\u2f0a\u4e86\u65b0\u7684\u5f3a\u2f24\u52a8\u2f12\uff1a\u7269\u7406\u3002<\/span><\/p>\n


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\u2f08\u4eec\u53d1\u73b0\u63cf\u8ff0\u4e8c\u7ef4\u6709\u7406\u5171\u5f62\u91cf\u2f26\u573a\u8bba\u548c\u4efb\u610f\u7ef4\u62d3\u6251\u5e8f\u7684\u6570\u5b66\u8bed\u2f94\u4e5f\u662f\u8303\u7574\u5b66\u3002\u5f53\u7136\u8fd9\u4e5f\u6ca1\u6709\u4ec0\u4e48\u5947\u602a\u7684\uff0c\u4e86\u89e3\u8303\u7574\u5b66\u7684\u2f08\u90fd\u4e0d\u4f1a\u60ca\u8bb6\u3002\u56e0\u4e3a\u8303\u7574\u5b66\u5e26\u6765\u7684\u53d8\u2fb0\u662f\u5982\u6b64\u5e95\u5c42\uff0c\u5b83\u4ece\u6839\u672c\u4e0a\u6539\u53d8\u4e86\u6211\u4eec\u770b\u5f85\u6570\u5b66<\/span>\uff08\u751a\u2f84\u662f\u5176\u4ed6\u5b66\u79d1\uff09<\/span>\u7684\u57fa\u672c\u8303\u5f0f\u3002\u2f7d\u4e0d\u4e86\u89e3\u8303\u7574\u5b66\u7684\u2f08\u53ef\u80fd\u4f1a\u5bf9\u8fd9\u53e5\u8bdd\u6709\u5f88\u2f24\u7684\u62b5\u89e6\u3002\u8fd9\u4e2a\u4e5f\u662f\u6b63\u5e38\u7684\uff0c\u6ca1\u6709\u771f\u6b63\u4e86\u89e3\u8303\u7574\u5b66\uff0c\u4f1a\u5f88\u96be\u60f3\u8c61\u8303\u7574\u5b66\u662f\u53ef\u80fd\u7684\uff0c\u4e5f\u8bb8\u770b\u5b8c\u6b64\u2f42\uff0c\u4f60\u7684\u62b5\u89e6\u4f1a\u7a0d\u7a0d\u51cf\u5c11\u2f00\u4e9b\u3002<\/span><\/p>\n


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\u6211\u8ba4\u4e3a\u8303\u7574\u5b66\u662f\u7ee7\u2f5c\u987f\u7684\u5fae\u79ef\u5206\u2fb0\u547d\u4e4b\u540e\uff0c\u2f1c\u2f00\u6b21\u8bed\u2f94\u7684\u2fb0\u547d\uff0c\u5176\u5b9e\u8303\u7574\u5b66\u672c\u8eab\u5c31\u662f\u2f00\u4e2a\u65b0\u7684\u5fae\u79ef\u5206\u3002\u5979\u7684\u2f12\u91cf\u4f53\u73b0\u5728\u8bb8\u591a\u2f45\u2faf\uff0c\u2f50\u5982\uff1a\u2f00\u4e2a\u7b80\u5355\u7684\u8303\u7574\u5b66\u7684\u516c\u5f0f\u5c31\u53ef\u4ee5\u5b8c\u6210\u2f00\u4e2a\u590d\u6742\u7684\u91cf\u2f26\u573a\u8bba\u7684\u6784\u9020\uff0c\u6216\u540c\u65f6\u8ba1\u7b97\u2f46\u7a77\u591a\u91cf\u2f26\u573a\u8bba\u7684\u878d\u5408 <\/span>(fusion)<\/span> \uff1b\u5f88\u591a\u590d\u6742\u7684\u7269\u7406\u548c\u6570\u5b66\u7ed3\u6784\u2f83\u7136\u2f7d\u7136\u5c31\u662f\u8303\u7574\u5b66\u7684\uff1b\u66f4\u91cd\u8981\u7684\u662f\uff0c\u5f88\u591a\u8d85\u8d8a\u96c6\u5408\u8bba\u7684\u6570\u5b66\u6216\u7269\u7406\u4e8b\u5b9e\u53ea\u80fd\u5728\u8303\u7574\u5b66\u7684\u610f\u4e49\u4e0b\u9648\u8ff0\u548c\u7406\u89e3\u3002\u4ec5\u4ec5\u662f\u6700\u540e\u2f00\u6761\u5c31\u544a\u8bc9\u6211\u4eec\uff0c\u6709\u2f00\u4e2a\u96c6\u5408\u8bba\u4e4b\u5916\u7684\u6570\u5b66\u65b0\u2f24\u9646\u7b49\u6211\u4eec\u53bb\u53d1\u73b0\u3001\u53bb\u63a2\u7d22\u3002\u73b0\u5728\u8fd8\u6709\u66f4\u75af\u72c2\u7684\u731c\u6d4b\uff0c\u90a3\u5c31\u662f\u8303\u7574\u5b66\u662f\u591a\u4f53\u91cf\u2f26\u7ea0\u7f20\u548c\u91cf\u2f26\u5f15\u2f12\u7684\u57fa\u7840\u3002<\/span><\/p>\n


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\u8303\u7574\u5b66\u7684\u53d8\u2fb0\u662f\u5982\u6b64\u5e95\u5c42\uff0c\u5b83\u4f1a\u6beb\u2f46\u7591\u95ee\u5730\u5728\u2f0f\u4e4e\u6240\u6709\u79d1\u5b66\u9886\u57df\u53d1\u6325\u4f5c\u2f64\uff0c\u5305\u62ec\u903b\u8f91\u5b66\u3001\u6570\u5b66\u3001\u7269\u7406\u3001\u8ba1\u7b97\u673a\u79d1\u5b66\u3001\u8bed\u2f94\u5b66\u3001\u793e\u4f1a\u5b66\u3001\u7ecf\u6d4e\u5b66\u7b49\u7b49\u3002\u6240\u4ee5\u8ba9\u66f4\u591a\u6570\u5b66\u4ee5\u5916\u7684\u2f08\u4e86\u89e3\u5b83\u662f\u6709\u610f\u4e49\u7684\u3002\u672c\u2f42\u5c31\u629b\u7816\u5f15\u2f5f\u5730\u8bb2\u8bb2\uff0c\u8fd9\u4e2a\u53d8\u2fb0\u662f\u5982\u4f55\u7684\u5e95\u5c42\uff0c\u5e95\u5c42\u5230\u9700\u8981\u6211\u4eec\u4e0d\u65ad\u5730\u56de\u5f52\uff0c\u76f4\u5230\u6bcf\u2f00\u4e2a\u2f08\u5f00\u59cb\u6570\u5b66\u542f\u8499\u7684\u90a3\u2f00\u523b\u3002<\/span><\/p>\n


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\u6211\u76f8\u4fe1\u6211\u4eec\u6bcf\u2f00\u4e2a\u2f08\u7684\u6570\u5b66\u6559\u80b2\u90fd\u662f\u4ece1+1=2\u5f00\u59cb\u7684\uff0c\u4ece\u90a3\u2f00\u523b\u5f00\u59cb\uff0c\u6211\u4eec\u5c31\u5f00\u542f\u4e86\u2f00\u573a\u201c\u53bb\u8303\u7574\u5316\u201d\u7684\u62bd\u8c61\u6570\u5b66\u4e4b\u65c5\uff0c\u2f7d\u8303\u7574\u5b66\u5219\u662f\u2f00\u573a\u56de\u5f52\u3002<\/span><\/p>\n


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\u521b\u9020\u529b\u7684\u6765\u6e90\u662f\u5929\u771f\u3002<\/span><\/section>\n
\u2014\u2014Alexander Grothendieck<\/span><\/section>\n<\/section>\n<\/blockquote>\n


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\u6211\u5e0c\u671b\u2f24\u5bb6\u548c\u6211\u2f00\u8d77\u56de\u5230\u5b66\u9f84\u524d\u2f09\u7ae5\u7684\u72b6\u6001\u3002\u53ea\u6709\u8fd9\u6837\u4f60\u624d\u80fd\u770b\u6e05\u95ee\u9898\u7684\u672c\u8d28\u3002<\/span><\/p>\n


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1+1=2 \u662f\u5f88\u96be\u7406\u89e3\u7684\u3002\u6211\u4eec\u771f\u6b63\u7406\u89e3\u4e86\u5417\uff1f\u4e5f\u8bb8\u4f60\u89c9\u5f97\u6ca1\u6709\u4ec0\u4e48\u56f0\u96be\uff0c\u4f46\u662f\u53ea\u6709\u5f53\u4f60\u7ed9\u2f00\u4e2a\u4ece\u6765\u6ca1\u6709\u542c\u8fc7 1+1=2 \u7684\u5b66\u9f84\u524d\u2f09\u7ae5\u89e3\u91ca\u7684\u65f6\u5019\uff0c\u4f60\u624d\u80fd\u660e\u2f69\u8fd9\u4e2a\u95ee\u9898\u6709\u591a\u4e48\u56f0\u96be\u3002<\/span><\/p>\n


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\u7b2c\u2f00\u4e2a\u96be\u70b9\u662f\uff1a\u4ec0\u4e48\u662f\u201c1\u201d\uff1f<\/span><\/p>\n

\u7b2c\u2f06\u4e2a\u96be\u70b9\u662f\uff1a\u4ec0\u4e48\u662f\u201c\uff0b\u201d\uff1f<\/span><\/p>\n

\u7b2c\u4e09\u4e2a\u96be\u70b9\u662f\uff1a\u4ec0\u4e48\u662f\u201c\uff1d\u201d\uff1f<\/span><\/p>\n

\u7b2c\u56db\u4e2a\u96be\u70b9\u662f\uff1a\u4ec0\u4e48\u662f\u201c2\u201d\uff1f<\/span><\/p>\n


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\u4ec0\u4e48\u662f\u201c1\u201d\uff1f\u4f60\u786e\u5b9a\u81ea\u5df1\u77e5\u9053\u4ec0\u4e48\u662f\u201c1\u201d\u5417\uff1f\u4f60\u2ec5\u8fc71\u5417\uff1f\u2f29\u670b\u53cb\u4e0d\u77e5\u9053\u4ec0\u4e48\u662f\u201c1\u201d \u3002\u4e3a\u4e86\u8ba9\u5c0f\u670b\u53cb\u7406\u89e3\u6570\u5b57\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u2f7c\u5e08\u7684\u6559\u6cd5\u662f\u2f64\u5b9e\u7269\uff0c\u2f50\u5982\u2f64\u5e26\u78c1\u94c1\u7684\u2f29\u732a\u3001\u2f29\u9e2d\u3001\u82f9\u679c\u3001\u2fb9\u8549\u7b49\u7b49\uff0c\u628a\u5b83\u4eec\u5438\u5230\u5e26\u2fa6\u5c5e\u7684\u2fca\u677f\u4e0a\u3002\u771f\u5b9e\u53ef\u2ec5\u7684\u4e1c\u2ec4\u624d\u662f\u6211\u4eec\u5bf9\u5b58\u5728\u7684\u57fa\u672c\u4f53\u9a8c\uff0c\u5176\u4ed6\u90fd\u4e0d\u592a\u53ef\u9760\u3002<\/span><\/p>\n


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\u8ba9\u6211\u2f64\u7b26\u53f7 O \u6765\u8868\u793a\u82f9\u679c\uff0cJ \u6765\u8868\u793a\u2fb9\u8549\u3002\u6211\u4eec\u518d\u628a\u2f0f\u4e2a\u82f9\u679c\u653e\u5230\u2f00\u8d77\uff0c\u4e8e\u662f\u2fca\u677f\u4e0a\u51fa\u73b0\u4e86\u5982\u4e0b\u516c\u5f0f\uff1a<\/span><\/p>\n


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O+O=OO,   \uff081\uff09<\/span><\/p>\n


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\u597d\u5427\u3002\u6211\u4eec\u2ec5\u8fc7\u82f9\u679c\uff0c\u6240\u4ee5O\u6ca1\u4ec0\u4e48\u95ee\u9898\u3002\u4f46\u4ec0\u4e48\u662f\u201c\uff0b\u201d\uff1f\u4ec0\u4e48\u662f\u201c\uff1d\u201d\u5462\uff1f<\/span><\/p>\n


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<\/p>\n

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<\/p>\n

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<\/p>\n

<\/p>\n


<\/p>\n

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<\/p>\n

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<\/p>\n

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<\/p>\n

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<\/p>\n

\u6709\u4e24\u79cd\u4e0d\u2f00\u6837\u7684\u201c\u2f00\u6837\u201d\u8fd8\u662f\u2f00\u6837\u5417\uff1f<\/span><\/strong><\/span><\/p>\n


<\/p>\n

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<\/p>\n

\n
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\n
\n
\n
\n

\u4e3a\u90a3\u4e9b\u6709\u7ebf\u6027\u4ee3\u6570\u57fa\u7840\u7684\u8bfb\u8005\u52a0\u2f00\u6bb5\uff1a<\/span><\/p>\n

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<\/p>\n

\u6211\u4eec\u6ce8\u610f\u5230\u4e0a\u2f42\u53cd\u590d\u51fa\u73b0\u4e86\u201c\u2f83\u7136\u201d\u8fd9\u4e2a\u8bcd\u3002\u2f7d\u8303\u7574\u5b66\u7684\u8d77\u6e90\uff0c\u5c31\u662fEllenberg-Mac Lane\u8bd5\u56fe\u5b9a\u4e49\u4ec0\u4e48\u662f\u201c\u2f83\u7136\u201d\uff0c\u7531\u6b64\u5f15\u53d1\u4e86\u201c\u2f83\u7136\u53d8\u6362\u201d<\/span> (natural transformation) <\/span>\u8fd9\u4e2a\u6982\u5ff5\uff0c\u4e3a\u4e86\u5b9a\u4e49\u201c\u2f83\u7136\u53d8\u6362\u201d\uff0c\u9700\u8981\u5f15\u2f0a\u201c\u51fd\u2f26\u201c<\/span>\uff08functor\uff09<\/span>\u7684\u6982\u5ff5\uff0c\u4e3a\u4e86\u5b9a\u4e49\u51fd\u2f26\uff0c\u2f1c\u9700\u8981\u5f15\u2f0a\u201c\u8303\u7574\u201d<\/span>\uff08category\uff09<\/span>\u8fd9\u4e2a\u6982\u5ff5\u3002<\/span><\/p>\n


<\/p>\n

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<\/p>\n

\u6211\u4eec\u60f3\u8bf4\uff0c\u4eceOO\u62bd\u8c61\u51fa\u6765\u2f00\u4e2a\u201c2\u201d\u7684\u6982\u5ff5\u5176\u5b9e\u662f\u2fae\u5e38\u56f0\u96be\u7684\uff0c\u2f7d\u4e14\u5f80\u5f80\u9700\u8981\u5f88\u66b4\u2f12\u7684\u505a\u6cd5\u3002\u2f7c\u5e08\u5728\u5f15\u2f0a\u201c2\u201d\u4e4b\u524d\uff0c\u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u8fd8\u4f1a\u518d\u653e\u4e24\u4e2a\u2fb9\u8549\u3002\u6211\u4eec\u59d1\u4e14\u2f64J\u6765\u4ee3\u8868\u2fb9\u8549\u3002\u4e8e\u662f\u2fca\u677f\u4e0a\u2f1c\u51fa\u73b0\u4e86\u5982\u4e0b\u516c\u5f0f\uff1a<\/span><\/p>\n


<\/p>\n

J + J = JJ\uff0c   (2)<\/span><\/p>\n


<\/p>\n

\u4f46\u662f\u540c\u6837\u7684\u95ee\u9898\u4ecd\u7136\u4f1a\u4ee4\u6211\u4eec\u70e6\u607c\u3002\u66f4\u52a0\u4ee4\u2f08\u56f0\u60d1\u7684\u662f\u2f7c\u5e08\u6709\u7684\u65f6\u5019\u8fd8\u8981\u5728\u2f9b\u5411\u201c2\u201d\u7684\u8def\u4e0a\u505a\u66f4\u591a\u8ba9\u6211\u4eec\u56f0\u60d1\u7684\u4e8b\u60c5\uff0c\u2f50\u5982\u4e3a\u4e86\u786c\u8bf4\u8fd9\u4e9b\u90fd\u662f\u201c2\u201d\uff0c\u8fd8\u53ef\u80fd\u6709\u8fd9\u6837\u7684\u516c\u5f0f\u51fa\u73b0\uff1a<\/span><\/p>\n


<\/p>\n

OO=JJ.<\/span><\/p>\n


<\/p>\n

\u8fd9\u6837\u4e0d\u5728\u2f00\u4e2a\u201c\u8303\u7574\u201d\u80fd\u201c\u2f00\u6837\u201d\u5417\uff1f\u751a\u2f84\u2f00\u4e0d\u7559\u795e\uff0c\u2f50\u5982\u82f9\u679c\u4e0d\u591f\u2f64\u4e86\uff0c\u53ef\u80fd\u4e34\u65f6\u8fd8\u4f1a\u51fa\u73b0\u4e0b\u2faf\u7684\u516c\u5f0f\uff1a<\/span><\/p>\n


<\/p>\n

O + J = J + J.<\/span><\/p>\n


<\/p>\n

\u75af\u6389\u4e86\uff0c\u82f9\u679c\u548c\u2fb9\u8549\u80fd\u52a0\u5417\uff1f\u82f9\u679c\u548c\u2fb9\u8549\u4e0d\u5728\u2f00\u4e2a\u201c\u8303\u7574\u201d\u600e\u4e48\u80fd\u52a0\u5462\uff1f\u4e8b\u5b9e\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u8bf4\u2f00\u4e2a\u82f9\u679c\u662f\u201c1\u201d\uff0c\u2f00\u4e2a\u2fb9\u8549\u4e5f\u662f\u201c1\u201d\uff0c\u5b83\u4eec\u90fd\u662f\u201c1\u201d\u7684\u4ee3\u8868\uff0c\u4f46\u662f\u4ece\u8fd9\u4e9b\u53ef\u4ee5\u4f5c\u4e3a\u201c1\u201d\u7684\u4ee3\u8868\u4e2d\u62bd\u8c61\u51fa\u6765\u201c1\u201d\u8fd9\u6837\u7684\u6982\u5ff5\u662f\u2fae\u5e38\u56f0\u96be\u7684\u3002\u4e5f\u8bb8\u90a3\u4e9b\u8fde1+1=2\u90fd\u542c\u4e0d\u61c2\u7684\u5b69\u2f26\u4e0d\u662f\u7b28\uff0c\u2f7d\u662f\u628a\u63e1\u4f4f\u4e86\u2f00\u4e9b\u6df1\u523b\u548c\u672c\u6e90\u7684\u4e1c\u2ec4\u3002<\/span><\/p>\n


<\/p>\n

\u6211\u4eec\u6765\u770b\u770b\u8303\u7574\u5b66\u600e\u4e48\u89e3\u8bfb1+1\u3002<\/span><\/p>\n

\n
\n


<\/p>\n


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

\n
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<\/section>\n

<\/section>\n
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<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

3. \u8303\u7574\u5b66\u7684\u89c2\u70b9\uff1a\u4e07\u6709\u6027\u8d28<\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

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


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

\u8303\u7574\u5b66\u7684\u89c2\u70b9\u5c31\u548c\u6211\u4eec\u6700\u5929\u771f\u7684\u770b\u6cd5\u2f00\u6837\uff0c\u2f00\u4e2a\u82f9\u679c\u662f\u201c1\u201d\uff0c\u2f00\u4e2a\u2fb9\u8549\u4e5f\u662f\u201c1\u201d\u3002\u5b83\u4eec\u90fd\u662f\u201c1\u201d\u7684\u4ee3\u8868\u3002\u65e2\u7136\u53ea\u662f\u4ee3\u8868\uff0c\u662f\u4e0d\u662f\u8bf4\u5b83\u4eec\u90fd\u8fd8\u4e0d\u662f\u201c1\u201d\uff1f\u90a3\u4e48\u5230\u5e95\u4ec0\u4e48\u662f\u201c1\u201d\u5462\uff1f
<\/span><\/p>\n


<\/p>\n

\u201c1\u201d\u5e94\u8be5\u53cd\u6620\u51fa\u6765\u6240\u6709\u8fd9\u4e9b\u201c1\u201d\u7684\u4e0d\u540c\u4ee3\u8868\u6240\u5177\u6709\u7684\u201c\u5171\u6709\u6027\u8d28\u201d\u3002\u6570\u5b66\u5bb6\u7ed9\u8fd9\u4e2a\u201c\u5171\u6709\u6027\u8d28\u201d\u8d77\u4e86\u2f00\u4e2a\u6b63\u5f0f\u7684\u540d\u5b57\u53eb\u201c\u4e07\u6709\u6027\u8d28\u201d<\/span> (universal property)<\/span> \u3002\u5982\u4f55\u5199\u4e0b\u8fd9\u4e9b\u201c1\u201d\u7684\u4e0d\u540c\u4ee3\u8868\u7684\u5171\u6709\u6027\u8d28\u5462\uff1f<\/span><\/p>\n


<\/p>\n

\u8303\u7574\u5b66\u63d0\u4f9b\u4e86\u2f00\u79cd\u5168\u65b0\u7684\u89c6\u2ec6\u3002\u4e0d\u8981\u2f64\u201c\u2f00\u4e2a\u7814\u7a76\u5bf9\u8c61\u201d\u2fa5\u2faf\u201c\u6709\u4ec0\u4e48\u4e1c\u2ec4\u201d\u8fd9\u6837\u2fae\u5e38\u96c6\u5408\u8bba\u6216\u8fd8\u539f\u8bba\u7684\u2f45\u5f0f\u53bb\u770b\u95ee\u9898\uff0c\u2f7d\u8981\u4ee5\u5bf9\u8c61\u548c\u5176\u4ed6\u5bf9\u8c61\u7684\u76f8\u4e92\u5173\u7cfb\u7684\u2f45\u5f0f\u6765\u4e86\u89e3\u2f00\u4e2a\u5bf9\u8c61\u3002\u8fd9\u4e2a\u2f45\u5f0f\u5176\u5b9e\u662f\u6211\u4eec\u7406\u89e3\u4e16\u754c\u66f4\u6839\u672c\u7684\u2f45\u6cd5\uff0c\u2f50\u5982\u4f60\u60f3\u4e86\u89e3\u2f00\u4e2a\u672a\u77e5\u7684\u201c\u5b58\u5728\u201d<\/span>\uff08\u5982\u7c92\u2f26\u3001\u6750\u6599\u7b49\uff09<\/span>\uff0c\u4f60\u600e\u4e48\u529e\uff1f\u4f60\u4f1a\u2f64\u4f60\u719f\u6089\u7684\u4e1c\u2ec4\u6253\u8fdb\u53bb\u53bb\u770b\u770b\u4f1a\u6d4b\u91cf\u51fa\u6765\u4ec0\u4e48\u3002\u7269\u7406\u5b66\u5bb6\u4f1a\u6d4b\u2f00\u4e2a\u65b0\u6750\u6599\u7684\u53d1\u5149\u8c31\u548c\u5438\u6536\u8c31\uff0c\u6253X\u5149\u8fdb\u53bb\u770b\u770bX\u5c04\u7ebf\u884d\u5c04\uff1b\u6570\u5b66\u5bb6\u4f1a\u628a\u7403\u2faf\u6254\u8fdb\u2f00\u4e2a\u672a\u77e5\u7a7a\u95f4\u6765\u6d4b\u91cf\uff0c\u6216\u770b\u770b\u80fd\u4e0d\u80fd\u8ba9\u2f00\u4e2a\u7fa4\u4f5c\u2f64\u4e0a\u53bb\uff0c\u7b49\u7b49\u3002\u2fbc\u80fd\u52a0\u901f\u5668\u7684\u4e91\u5ba4\u2fa5\u2faf\u6d4b\u7684\u4e0d\u662f\u7c92\u2f26\u7684\u8f68\u8ff9\uff0c\u2f7d\u662f\u7c92\u2f26\u548c\u5176\u4ed6\u4e1c\u2ec4\u76f8\u4e92\u4f5c\u2f64\u7684\u8f68\u8ff9\u3002\u6ca1\u6709\u76f8\u4e92\u4f5c\u2f64\uff0c\u6d4b\u91cf\u4e5f\u2f46\u4ece\u8c08\u8d77\u3002\u53ef\u4ee5\u2f17\u5206\u5b89\u5168\u5730\u8bf4\uff1a<\/span><\/p>\n


<\/p>\n

\u8fd9\u4e2a\u4e16\u754c\u4e0a\u6ca1\u6709\u2f50\u76f8\u4e92\u5173\u7cfb\u6216\u76f8\u4e92\u4f5c\u2f64\u66f4\u57fa\u672c\u7684\u5b58\u5728\u3002<\/span><\/p>\n


<\/p>\n

\u65e2\u7136\u5982\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u2f64\u76f8\u4e92\u5173\u7cfb\u6765\u5b9a\u4e49\u4ec0\u4e48\u662f\u201c1\u201d\u3002<\/span><\/p>\n


<\/p>\n

\u6211\u4eec\u5148\u56de\u987e\u2f00\u4e2a\u6982\u5ff5\uff1a\u96c6\u5408\u4e4b\u95f4\u7684\u6620\u5c04 <\/span>(a map)<\/span> \u3002\u96c6\u5408\u5c31\u662f\u2f00\u5806\u5143\u7d20\u7684\u201c\u96c6\u5408\u201d\uff0c\u5475\u5475\u3002\u4e0d\u8fc7\u503c\u5f97\u6307\u51fa\u7684\u662f\uff0c\u7a7a\u96c6\u4e5f\u662f\u2f00\u4e2a\u96c6\u5408\uff0c\u5c31\u662f\u2f00\u4e2a\u6ca1\u6709\u5143\u7d20\u7684\u96c6\u5408\u3002\u90a3\u4e48\u4ec0\u4e48\u662f\u4e24\u4e2a\u96c6\u5408A\u548cB\u4e4b\u95f4\u7684\u6620\u5c04\u5462\uff1f\u2f50\u5982\u8003\u8651\u4e24\u4e2a\u96c6\u5408X\uff1d{a,b}, Y={1,2,3}, \u2f00\u4e2a\u4eceX\u5230Y\u7684\u6620\u5c04\uff0c\u8bb0\u6210<\/span><\/p>\n

<\/p>\n

\u6216<\/span><\/p>\n

<\/p>\n


<\/p>\n

\u8fd9\u5176\u5b9e\u5c31\u662f\u2f00\u4e2a\u5206\u914d\u89c4\u5219\uff1a\u7ed9X\u4e2d\u7684\u6bcf\u2f00\u4e2a\u5143\u7d20\u5206\u914d\u552f\u2f00\u2f00\u4e2aY\u4e2d\u7684\u5143\u7d20\u3002\u2f50\u5982\uff0cf(a)=1, f(b)=1 \u5c31\u662f\u2f00\u4e2a\u5408\u7406\u7684\u6620\u5c04, g(a)=2, g(b)=3 \u4e5f\u662f\u2f00\u4e2a\u6620\u5c04\u3002\u4f46\u662f\u4e0d\u80fd\u7ed9a\u5206\u914d\u4e24\u4e2aY\u4e2d\u7684\u5143\u7d20\uff01\u5982\u679c\u96c6\u5408X\u2fa5\u2faf\u6ca1\u6709\u5143\u7d20\uff08\u7a7a\u96c6\uff09\uff0c\u7b49\u4e8e\u5206\u914d\u89c4\u5219\u2f83\u52a8\u5b9a\u4e49\u597d\u4e86\uff0c\u8fd9\u4e2a\u4ec0\u4e48\u90fd\u4e0d\u9700\u8981\u5206\u914d\u7684\u5206\u914d\u89c4\u5219\u5c31\u53eb\u7a7a\u6620\u5c04\u3002<\/span><\/p>\n


<\/p>\n

\u6709\u4e86\u8fd9\u4e9b\u51c6\u5907\uff0c\u6211\u4eec\u53ef\u4ee5\u7ed9\u51fa\u2f00\u4e0b\u5b9a\u4e49\u3002<\/span><\/p>\n


<\/p>\n

\u5b9a\u4e49\uff1a1 \u5c31\u662f\u8fd9\u6837\u2f00\u4e2a\u96c6\u5408\uff0c\u4efb\u4f55\u2f00\u4e2a\u96c6\u5408\u5230\u5b83\u90fd\u5b58\u5728\u4e14\u6709\u552f\u2f00\u2f00\u4e2a\u6620\u5c04[1]\u3002\u6211\u4eec\u6709\u2f00\u4e2a\u7b80\u6d01\u7684\u56fe\u6765\u8bb0\u5f55\u8fd9\u4e2a\u5b9a\u4e49\uff1a\u5bf9\u4efb\u610f\u96c6\u5408X\uff0c\u6211\u4eec\u6709<\/span><\/p>\n

<\/p>\n


<\/p>\n

\u8fd9\u2fa5\u201c<\/span>\u201d\u662f\u6307\u201c\u5b58\u5728\u201d\uff0c\u201c\uff01\u201d\u662f\u6307\u201c\u552f\u2f00\u201d[1]\u3002\u53e6\u5916\u8981\u6ce8\u610f\uff0c\u5b9a\u4e49\u4e2d\u201c\u5bf9\u4efb\u610f\u96c6\u5408X\u201d\u4e5f\u2fae\u5e38\u91cd\u8981\uff01\u4e0d\u662f\u5bf9\u2f00\u4e2a\u7279\u522b\u7684\u96c6\u5408\uff0c\u2f7d\u662f\u6240\u6709\u96c6\u5408\uff01<\/span><\/p>\n


<\/p>\n

\u2f24\u5bb6\u770b\u5230\u6ca1\u6709\uff0c\u8fd9\u4e2a\u5b9a\u4e49\u2fa5\u2faf\u2f64\u5230\u4e86\u201c1\u201d\u548c\u6240\u6709\u96c6\u5408\u7684\u5173\u7cfb\uff0c\u8fd9\u4ef6\u4e8b\u76f8\u5f53\u91cd\u8981\u3002\u4e0d\u8fc7\u7b2c\u2f00\u6b21\u770b\u5230\u8fd9\u4e2a\u8bfb\u8005\u53ef\u80fd\u66f4\u5173\u2f3c\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u8fd9\u662f\u201c1\u201d\u7684\u2f00\u4e2a\u5408\u7406\u7684\u5b9a\u4e49\u5462\uff1f\u6211\u4eec\u6765\u770b\u770b\uff0c\u2f00\u4e2a\u82f9\u679c\u7684\u96c6\u5408\u6ee1\u4e0d\u6ee1\u2f9c\u8fd9\u4e2a\u5b9a\u4e49\uff1f\u2f00\u4e2a\u2fb9\u8549\u7684\u96c6\u5408\u5462\uff1f\u6216\u8005\uff0c\u96f6\u4e2a\u6216\u4e09\u4e2a\u2fb9\u8549\u5462\uff0c\u2f1c\u6216\u8005\u6240\u6709\u4e2d\u56fd\u2f08\u7684\u96c6\u5408\uff1f<\/span><\/p>\n


<\/p>\n

\u5982\u679c\u4f60\u613f\u610f\u5c1d\u8bd5\uff0c\u4f60\u5f88\u5feb\u4f1a\u53d1\u73b0\uff0c\u96f6\u4e2a\u2fb9\u8549\u662f\u4e0d\u2f8f\u7684\uff0c\u56e0\u4e3a\u5b83\u7834\u574f\u4e86\u5b9a\u4e49\u4e2d\u6620\u5c04\u201c\u5b58\u5728\u6027\u201d\u6761\u4ef6\u3002\u201c\u4e09\u4e2a\u2fb9\u8549\u201d\u4e5f\u662f\u4e0d\u2f8f\u7684\uff0c\u56e0\u4e3a\u5b83\u7834\u574f\u4e86\u6620\u5c04\u7684\u201c\u552f\u2f00\u6027\u201d\u3002\u4ec0\u4e48\u96c6\u5408\u53ef\u4ee5\u5462\uff1f\u5c31\u662f\u90a3\u4e9b\u53ea\u6709\u2f00\u4e2a\u5143\u7d20\u7684\u96c6\u5408\uff0c\u2f50\u5982\u2f00\u4e2a\u82f9\u679c\u7684\u96c6\u5408\u3001\u2f00\u4e2a\u2fb9\u8549\u7684\u96c6\u5408\u3001\u2f00\u4e2a\u9e21\u86cb\u7684\u96c6\u5408\u3001\u2f00\u4e2a\u2f08\u7684\u96c6\u5408\uff0c\u7b49\u7b49\uff0c\u5b83\u4eec\u53ef\u4ee5\u540c\u65f6\u4fdd\u8bc1\u5b58\u5728\u6027\u548c\u552f\u2f00\u6027\u3002<\/span><\/p>\n


<\/p>\n

\u6240\u4ee5\u8fd9\u6837\u5b9a\u4e49\u7684\u201c1\u201d\u4e0d\u552f\u2f00\uff0c\u8fd9\u4e2a\u597d\u50cf\u662f\u2f00\u4e2a\u7f3a\u9677\uff0c\u4f46\u662f\u5999\u7684\u5730\u2f45\u662f\uff0c\u6240\u6709\u53ef\u80fd\u7684\u201c1\u201d\u90fd\u6709\u4e14\u4ec5\u6709\u2f00\u79cd\u2f45\u5f0f\u4e92\u76f8\u5bf9\u5e94\u8d77\u6765\uff0c\u8fd9\u662f\u7531\u201c\u5b58\u5728\u6027\u201d\u548c\u201c\u552f\u2f00\u6027\u201d\u51b3\u5b9a\u7684\uff0c\u548c\u201c1\u201d\u7684\u5177\u4f53\u5185\u5bb9\u2f46\u5173\u3002\u5c31\u50cf\u6559\u2f29\u670b\u53cb\u65f6\uff0c\u53ef\u4ee5\u2f64\u2f00\u4e2a\u82f9\u679c\u4ee3\u88681\uff0c\u4e5f\u53ef\u4ee5\u2f64\u2f00\u4e2a\u2fb9\u8549\u4ee3\u88681\uff0c\u2f7d\u4e14\u6211\u4eec\u77e5\u9053\u5982\u4f55\u628a\u5b83\u4eec\u7b49\u540c\u8d77\u6765! \u4f60\u80fd\u76f8\u4fe1\u5417\uff0c\u5e7c\u2f09\u56ed\u867d\u7136\u5f88\u52aa\u2f12\u5730\u6559\u201c\u53bb\u8303\u7574\u5316\u201d\u7684\u6570\u5b66\uff0c\u4f46\u6559\u7684\u2f45\u6cd5\u662f\u2f46\u6cd5\u56de\u907f\u7684\u8303\u7574\u5b66\uff01\u56e0\u4e3a\u8fd9\u5c31\u662f\u5b83\u7684\u672c\u6765\u2faf\u2f6c\u3002<\/span><\/p>\n


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\u597d\u6781\u4e86\uff0c\u5982\u679c\u4f60\u8fd8\u80fd\u8ddf\u4e0a\u6211\uff0c\u6211\u4eec\u5c31\u518d\u6765\u2f00\u4e2a\u3002<\/span><\/p>\n


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\u5b9a\u4e49\uff1a0\u5c31\u662f\u8fd9\u6837\u2f00\u4e2a\u96c6\u5408\uff0c\u5b83\u5230\u4efb\u4f55\u2f00\u4e2a\u96c6\u5408\u90fd\u5b58\u5728\u4e14\u6709\u552f\u2f00\u2f00\u4e2a\u6620\u5c04\u3002\u5373\u5bf9\u4efb\u610f\u96c6\u5408X\uff0c\u6211\u4eec\u6709\uff1a<\/span><\/p>\n

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\u8fd9\u4e2a\u7559\u7ed9\u2f24\u5bb6\u505a\u7ec3\u4e60\u5427\u3002\u2014\u2014\u79d1\u666e\u8fd8\u8981\u7559\u4f5c\u4e1a\uff1f\u6ca1\u542c\u8bf4\u8fc7\uff0c\u5475\u5475\u3002\u4e0d\u8fc7\u60f3\u660e\u2f69\u4e86\u8fd9\u4e2a\u4e60\u9898\uff0c\u2ee2\u4e0a\u5c31\u798f\u5229\u4e86\uff0c\u53ef\u4ee5\u2f74\u2ee2\u53bb\u6467\u6b8b\u2f29\u670b\u53cb\u548c\u5979\u4eec\u7684\u2f57\u2e9f\u554a\u3002\u54c8\u54c8\u3002<\/span><\/p>\n


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4. \u8303\u7574\u5b66\u600e\u4e48\u89e3\u8bfb1+1\uff1f<\/strong><\/strong><\/p>\n<\/section>\n

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


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\u597d\u4e86\uff0c\u771f\u6b63\u7684\u6311\u6218\u6216\u6467\u6b8b\u6765\u4e86\uff0c\u6211\u4eec\u7ec8\u4e8e\u53ef\u4ee5\u770b\u770b\u4ec0\u4e48\u662f1+1\u4e86\u3002\u548c\u201c1\u201d\u2f00\u6837\uff0c\u201c1+1\u201d\u4e5f\u4f1a\u6709\u5f88\u591a\u4e0d\u540c\u7684\u4ee3\u8868\uff0c\u2f50\u5982\uff0c2\u4e2a\u82f9\u679c\u62162\u4e2a\u2fb9\u8549\uff0c\u7b49\u7b49\u3002\u90a3\u4e48\u201c1+1\u201d\u5e94\u8be5\u662f\u4ec0\u4e48\u5462\uff1f\u5e94\u8be5\u662f\u6240\u6709\u8fd9\u4e9b\u4ee3\u8868\u6240\u62e5\u6709\u7684\u5171\u6027\uff0c\u5373\u4e07\u6709\u6027\u8d28\u3002\u4e0b\u2faf\u6211\u4eec\u5c31\u6765\u63ed\u793a\u201c1+1\u201d\u7684\u4e07\u6709\u6027\u8d28\u3002\u6211\u5148\u64e6\u64e6\u6c57\u3002<\/span><\/p>\n


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\u5f00\u52a8\u8111\u7b4b\u7684\u65f6\u5019\u5230\u4e86<\/span><\/section>\n


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\u8bb21+1\u4e4b\u524d\uff0c\u8fd8\u9700\u8981\u2f00\u4e2a\u51c6\u5907\u3002\u4e24\u4e2a\u8fde\u5728\u2f00\u8d77\u7684\u6620\u5c04 <\/span> \u53ef\u4ee5\u5408\u6210\u4e3a\u2f00\u4e2a\u4eceX\u5230Z\u7684\u6620\u5c04\uff0c\u8bb0\u4e3a<\/span>\u3002\u2f50\u5982\uff1aX={1,2,3}, Y={a,b}, Z={A,B,C,D}, f(1)=f(2)=a, f(3)=b, g(a)=A, g(b)=B, \u90a3\u4e48\u6211\u4eec\u6709\uff1a<\/span>\uff0c<\/span>\u3002<\/span><\/p>\n


<\/p>\n

\u5b9a\u4e49\uff1a1+1 \u5c31\u662f\u8fd9\u6837\u2f00\u4e2a\u96c6\u5408\uff0c\u5b83\u2f83\u52a8\u9644\u5e26\u4e24\u4e2a\u6307\u5b9a\u7684\u4ece1\u6765\u7684\u6620\u5c04\uff1a<\/span><\/p>\n


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\u4ece\u2f7d\u5bf9\u4efb\u4f55\u9644\u5e26\u4e24\u4e2a\u4ece1\u6765\u7684\u6620\u5c04\u7684\u96c6\u5408X\uff1a<\/span><\/p>\n


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\u90fd\u5b58\u5728\u4e14\u6709\u552f\u2f00\u2f00\u4e2a\u6620\u5c04\uff1a <\/span>\uff0c\u4f7f\u5f97 <\/span>\uff0c<\/span>\u3002<\/span><\/p>\n


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<\/p>\n

\u8fd9\u5c31\u662f\u201c1+1\u201d\u7684\u4e07\u6709\u6027\u8d28\uff0c\u4e5f\u540c\u65f6\u662f\u5b83\u7684\u5b9a\u4e49\u3002\u5982\u679c\u6211\u4eec\u2f64\u56fe\u6765\u8bb0\u5f55\u4e0a\u2faf1+1\u7684\u5b9a\u4e49\u6216\u4e07\u6709\u6027\u8d28\uff0c\u90a3\u4e48\u5c31\u662f\u4e0b\u2faf\u8fd9\u4e2a\u88ab\u79f0\u4e3a\u201c\u4ea4\u6362\u56fe\u201d\u7684\u4e1c\u2ec4\u3002<\/span><\/p>\n


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\u6240\u8c13\u56fe\u662f\u4ea4\u6362\u7684\u610f\u601d\u5c31\u662f\uff1a <\/span>\uff0c<\/span>\u3002\u8fd9\u4e2a\u5b9a\u4e49\u4e3a\u4ec0\u4e48\u5c31\u662f\u6211\u4eec\u719f\u77e5\u76841+1\u5462\uff1f\u8fd9\u8fd8\u771f\u662f\u4e0d\u597d\u89e3\u91ca\u3002\u4e3b\u8981\u662f\u9700\u8981\u89e3\u91ca\u7684\u6709\u70b9\u2f09\u591a\uff0c\u80fd\u7559\u505a\u4e60\u9898\u5417\uff1f\u6211\u4eec\u4e0d\u662f\u8fd8\u6709\u5f88\u591a\u5f3a\u2f24\u7684\u8bfb\u8005\u5417\uff1f\u5c31\u770b\u4f60\u4eec\u7684\u4e86\u3002\u6211\u5df2\u7ecf\u88ab\u6467\u6b8b\u5f97\u4e0d\u2f8f\u4e86\uff0c\u9700\u8981\u4f11\u606f\uff0c\u4f11\u606f\u2f00\u4f1a\u2f09\u3002<\/span><\/p>\n


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<\/section>\n
\u4f11\u606f\uff0c\u4f11\u606f\u2f00\u4e0b<\/span><\/section>\n


<\/p>\n

\u6211\u60f3\u6307\u51fa\u7684\u662f\uff1a<\/span><\/p>\n


<\/p>\n

\uff081\uff09\u867d\u7136\u6211\u4eec\u6ca1\u6709\u89c4\u5b9aa\u548cb\u5fc5\u987b\u662f\u4ec0\u4e48\uff0c\u4f46\u662f\u4e07\u6709\u6027\u8d28\u5bfc\u81f4\u4e86a\u548cb\u4e0d\u80fd\u4efb\u610f\u9009\u3002\u4e0d\u540c\u7684a \u548c b\u4e14\u6ee1\u2f9c\u4e07\u6709\u6027\u8d28\u7684\u9009\u62e9\u5c06\u88ab\u89c6\u4e3a\u4e0d\u540c\u76841+1\u7684\u4ee3\u8868\uff01\u4e5f\u5c31\u662f\u8bf4\u5b9a\u4e491+1\u9700\u8981\u6709\u4e09\u4e2a\u4e1c\u2ec4\uff1a(1+1, a, b)\u3002<\/span><\/p>\n


<\/p>\n

\uff082\uff09\u6ee1\u2f9c1+1\u7684\u5b9a\u4e49\u7684\u96c6\u5408\u662f\u4e0d\u552f\u2f00\u7684<\/span>\uff08\u90fd\u662f\u4ee3\u8868\uff09<\/span>\uff0c\u4f46\u662f\u5b58\u5728\u6027\u548c\u552f\u2f00\u6027\u4f7f\u5f97\uff0c\u5b83\u4eec\u4efb\u610f\u4e24\u4e2a\u4ee3\u8868\u4e4b\u95f4\u90fd\u6709\u552f\u2f00\u7684\u2f00\u79cd\u2f45\u5f0f\u2f00\u2f00\u5bf9\u5e94\u8d77\u6765\u3002\u8fd9\u2f00\u70b9\u76f8\u5f53\u7684\u91cd\u8981\uff0c\u4f46\u662f\u6211\u4e0d\u662f\u5f88\u60f3\u5c55\u5f00\u6765\u8bf4\uff0c\u53ef\u80fd\u4e5f\u9700\u8981\u8bfb\u8005\u4e2d\u7684\u8fbe\u2f08\u6765\u89e3\u8bfb\u3002<\/span><\/p>\n


<\/p>\n

\uff083\uff09\u4e07\u6709\u6027\u8d28\u7684\u53e6\u2f00\u5947\u5999\u7684\u5730\u2f45\u662f\uff0c\u5b83\u4e0d\u4f46\u5b9a\u4e49\u4e86\u6982\u5ff5\uff0c\u8fd8\u544a\u8bc9\u4f60\u5b83\u662f\u600e\u4e48\u2f64\u7684\uff0c\u5c31\u662f\u2f64\u6765\u6784\u9020\u90a3\u4e2a\u5b58\u5728\u552f\u4e00\u7684\u7bad\u5934\uff01\u2f7d\u4e14\u8fd9\u662f\u552f\u2f00\u7684\u2f64\u6cd5\uff01\u8fd9\u2f00\u70b9\u2f24\u6982\u6ca1\u6709\u2f64\u8fc7\u7684\u2f08\u662f\u5f88\u96be\u4f53\u4f1a\u7684\u3002\u8fd9\u4e2a\u96c6\u5b9a\u4e49\u4e0e\u5e94\u2f64\u4e8e\u2f00\u8eab\u7684\u7279\u70b9\u4e5f\u5f3a\u70c8\u8bc1\u660e\u4e86\u8fd9\u662f\u2f00\u4e2a\u597d\u7684\u5b9a\u4e49\u3002<\/span><\/p>\n


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\n
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\u4f5c\u4e1a1:  \u6211\u4eec\u5230\u5e95\u662f\u5b9a\u4e49\u4e862\u8fd8\u662f\u5b9a\u4e49\u4e86\u201c+\u201d\uff1f<\/span><\/p>\n

\u4f5c\u4e1a2\uff1a1 + 1 + 1 \u548b\u5b9a\u4e49\uff1f<\/span><\/p>\n

\u4f5c\u4e1a3\uff1a1 x 1\u548b\u5b9a\u4e49\uff1f\uff08\u63d0\u793a\uff1a\u628a\u5b9a\u4e491+1\u7684\u90a3\u4e2a\u56fe\u7684\u7bad\u5934\u90fd\u53cd\u8fc7\u6765\uff09\u3002<\/span><\/p>\n


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\u4f5c\u4e1a3\u76f8\u5f53\u6709\u610f\u601d\uff0c\u7bad\u5934\u90fd\u53cd\u8fc7\u6765\u5c31\u53ef\u4ee5\u4e86\uff1f\u5176\u5b9e\u8fd9\u4e5f\u5c31\u662f\u5728\u8bf4\uff0c\u4e58\u6cd5\u662f\u52a0\u6cd5\u7684\u5bf9\u5076\u6982\u5ff5\u3002\u8303\u7574\u5b66\u2f5c\u903c\u7684\u5730\u2f45\u662f\u8bf4\uff0c\u6570\u5b66\u2fa5\u7684\u6240\u6709\u6982\u5ff5\u90fd\u662f\u8fd9\u6837\u7684\uff01\u6570\u5b66\u2fa5\u7684\u6240\u6709\u6982\u5ff5\u53ea\u6709\u4e24\u79cd\uff1a\u2f00\u79cd\u53eb\u505a\u201c\u6781\u9650\u201d<\/span>\uff08\u2f50\u59821 x 1\uff09<\/span>\uff0c\u2f00\u79cd\u53eb\u201c\u4f59\u6781\u9650\u201d<\/span>\uff08\u2f50<\/span>\u59821+1\uff09<\/span>\u3002\u5176\u4ed6\u6ca1\u4e86\u3002\u5475\u5475\u3002\u6240\u4ee5\u8303\u7574\u5b66\u628a\u6570\u5b66\u4e2d\u7684\u6240\u6709\u6982\u5ff5\u90fd\u653e\u5728\u4e86\u2f00\u4e2a\u7edf\u2f00\u7684\u6846\u67b6\u2fa5\u2faf\u6765\u770b\u3002<\/span><\/p>\n


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\u4f17\u4f4d\u770b\u5b98\u53ef\u80fd\u8981\u7eb3\u95f7\uff0c\u8fd9\u4e2a\u600e\u4e48\u80fd\u53eb\u201c\u6781\u9650\u201d\u5462\uff1f\u6781\u9650\u4e0d\u662f\u2f00\u4e2a\u2f46\u9650\u7684<\/span>\uff08\u903c\u8fd1\uff09<\/span>\u8fc7\u7a0b\u5417\uff1f\u4e8b\u5b9e\u4e0a\uff0c\u2f24\u5bb6\u719f\u6089\u7684\u6240\u8c13\u201c\u6781\u9650\u201d\u4e0d\u8fc7\u662f\u2f64\u2f00\u4e2a\u6709\u2f46\u9650\u8282\u70b9\u7684\u4ea4\u6362\u56fe\uff0c\u5e76\u4ee5\u548c1+1 \u62161 x 1\u540c\u6837\u7684\u2f45\u5f0f\u6765\u5b9a\u4e49\u7684\u6982\u5ff5\u2f7d\u5df2\u3002\u2f50\u5982\uff0c\u5bf9\u4efb\u610f\u6784\u6210\u5982\u4e0b\u56fe\u7684\u5b9e\u6570X\uff0c\u6211\u4eec\u6709\uff1a<\/span><\/p>\n


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<\/p>\n

\u5176\u4e2d\u201c1\u201d\u662f\u901a\u5e38\u610f\u4e49\u4e0b\u7684\u2f00\u4e2a\u5b9e\u6570<\/span>\uff08\u4e0d\u8981\u7406\u89e3\u6210\u96c6\u5408\uff09<\/span>\uff0c\u7bad\u5934\u7684\u610f\u601d\u5c31\u662f\u201c\u2264 \u201d <\/span>(\u2f29\u4e8e\u7b49\u4e8e\uff09\uff08\u4e0d\u662f\u96c6\u5408\u4e4b\u95f4\u7684\u6620\u5c04\uff01\uff09<\/span>\u3002\u8fd9\u4e2a\u56fe\u8bf4\u7684\u662f\uff0c\u2f00\u4e2a\u5e8f\u5217\uff1a0.9, 0.99, 0.999, … \u7684\u6781\u9650\u662f1\u3002\u2f64\u8303\u7574\u5b66\u7684\u8bed\u2f94\u8bf4\u5c31\u662f1\u662f\u56fe\u8868 0.9\u21920.99\u21920.999\u2192… \u7684\u4f59\u6781\u9650\u3002<\/span><\/p>\n


<\/p>\n

\u600e\u4e48\u770b\u5462\uff1fX\u662f\u8fd9\u6837\u7684\u2f00\u4e2a\u5b9e\u6570\uff0c\u5e8f\u5217 0.9, 0.99, 0.999, …\u4e2d\u7684\u6bcf\u2f00\u4e2a\u6570\u5230X\u90fd\u6709\u2f00\u4e2a\u7bad\u5934\uff0c\u610f\u601d\u662f\u8bf4\uff0c\u5e8f\u5217 0.9, 0.99, 0.999, …\u4e2d\u7684\u6bcf\u2f00\u4e2a\u90fd \u2264 X \u3002\u2f7d1\u5c31\u662f\u8fd9\u6837\u7684\u2f00\u4e2a\u6570\uff0c\u2f7d\u4e14\u662f\u6700\u2f29\u7684\u90a3\u2f00\u4e2a\uff0c\u5bf9\u5427\uff1f<\/span><\/p>\n


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\u5f88\u591a\u2f08<\/span>\uff08\u5305\u62ec\u90e8\u5206\u6570\u5b66\u5bb6\uff09<\/span>\u90fd\u62b1\u6028\u8303\u7574\u5b66\u62bd\u8c61\u3002\u6211\u5e0c\u671b\u524d\u2faf\u7684\u8ba8\u8bba\u80fd\u591f\u5e2e\u52a9\u2f24\u5bb6\u610f\u8bc6\u5230\uff0c\u6211\u4eec\u7684\u601d\u8003\u5929\u2f63\u5c31\u662f\u8303\u7574\u5b66\u7684\uff0c\u2f7d\u5f15\u2f0a\u201c1\u201d\u548c\u201c2\u201d\u8fd9\u6837\u7684\u62bd\u8c61\u6982\u5ff5\u53cd\u2f7d\u662f\u201c\u53bb\u8303\u7574\u5316\u201d\u4e86\u3002\u6211\u4eec\u7684\u6570\u5b66\u6559\u80b2\u4ece\u2f00\u5f00\u59cb\u5c31\u662f\u201c\u53bb\u8303\u7574\u5316\u201d\uff0c\u2f7d\u901a\u5e38\u7684\u5fae\u79ef\u5206\u53ef\u4ee5\u770b\u4f5c\u662f\u201c\u53bb\u8303\u7574\u5316\u201d\u7684\u7ecf\u5178\u4e4b\u4f5c\u3002\u6700\u7ec8\u7684\u7ed3\u679c\u662f\uff0c\u6211\u4eec\u2f24\u591a\u6570\u2f08\u7b2c\u2f00\u6b21\u5b66\u201c\u8303\u7574\u5b66\u201d\u90fd\u4f1a\u89c9\u5f97\u597d\u201c\u62bd\u8c61\u201d\uff0c\u5475\u5475\u3002\u6709\u53ef\u80fd\u662f\u56e0\u4e3a \u201c\u53bb\u8303\u7574\u5316\u201d\u7684\u6570\u5b66\u6559\u80b2\u8ba9\u6211\u4eec\u53d8\u5f97\u5931\u53bb\u7ae5\u771f\u4e86\u3002<\/span><\/p>\n


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\u6211\u8bb0\u5f97\u6709\u2f00\u6b21\u5728\u505a\u6570\u5b66\u62a5\u544a\uff0c\u6709\u542c\u4f17\u62b1\u6028\u8303\u7574\u5b66\u592a\u62bd\u8c61\u4e86\u3002\u6211\u8bf4\uff0c\u62bd\u8c61\u662f\u2f00\u4e2a\u6ca1\u6709\u610f\u4e49\u7684\u6982\u5ff5\uff0c\u4e0d\u8fc7\u4f60\u6240\u8c13\u7684\u4e0d\u62bd\u8c61\u7684\u4e1c\u2ec4\u662f\u5565\uff1f\u4ed6\u56de\u7b54\u8bf4\u2f50\u5982\u4e0a\u540c\u8c03\u3002\u6211\u7684\u5929\uff0c\u4e0a\u540c\u8c03\u4e0d\u62bd\u8c61\uff1f\u597d\u5427\uff0c\u6211\u8010\u2f3c\u5730\u95ee\uff0c\u4e3a\u4ec0\u4e48\u4f60\u89c9\u5f97\u4e0a\u540c\u8c03\u4e0d\u62bd\u8c61\uff1f\u4ed6\u8bf4\uff0c\u56e0\u4e3a\u53ef\u4ee5\u7b97\u554a\u3002\u6211\u8bf4\uff0c\u539f\u6765\u53ef\u4ee5\u7b97\u5c31\u662f\u4e0d\u62bd\u8c61\u554a\uff0c\u8fd9\u6837\u7684\u8bdd\uff0c\u8303\u7574\u5b66\u4e5f\u4e0d\u62bd\u8c61\uff0c\u56e0\u4e3a\u5b83\u4e5f\u53ef\u4ee5\u7b97\u3002\u4f46\u662f\u8fd9\u4e5f\u4e0d\u91cd\u8981\uff0c\u56e0\u4e3a\u8fd9\u4e2a\u8bf4\u6cd5\u672c\u8eab\u5f88\u8352\u5510\uff0c\u5982\u679c\u4e0a\u540c\u8c03\u662f\u53ef\u4ee5\u7b97\u7684\u8bdd\uff0c\u603b\u6ca1\u67091+1=2\u66f4\u597d\u7b97\u5427\uff1f\u90a3\u4e48\u8bf7\u95ee\u4ec0\u4e48\u662f\u201c1\u201d\uff1f\u5e73\u65f6\u6211\u4eec\u5634\u4e0a\u8bf4\u7684\u6240\u8c13\u201c\u4e0d\u62bd\u8c61\u201d\u6216\u201c\u62bd\u8c61\u201d\uff0c\u5176\u5b9e\u5c31\u662f\u201c\u719f\u6089\u201d\u6216\u201c\u4e0d\u719f\u6089\u201d\u3002\u8303\u7574\u5b66\u4e4b\u6240\u4ee5\u663e\u5f97\u201c\u62bd\u8c61\u201d\uff0c\u5c31\u662f\u56e0\u4e3a\u6211\u4eec\u5728\u201c\u53bb\u8303\u7574\u5316\u201d\u7684\u8def\u4e0a\u2f9b\u4e86\u5f88\u8fdc\u4e86\uff0c\u60f3\u8981\u56de\u5f52\u4e5f\u6ca1\u6709\u90a3\u4e48\u5bb9\u6613\uff0c\u653e\u4e0b\u5305\u88b1\u662f\u5f88\u96be\u7684\u3002<\/span><\/p>\n


<\/span><\/p>\n

\u6211\u8bb0\u5f97\u6709\u2f00\u6b21\u548c\u7269\u7406\u5b66\u5bb6Michael Levin\u5403\u5348\u996d\uff0c\u4ed6\u8bf4\u4ed6\u82b1\u4e0d\u5c11\u65f6\u95f4\u770b\u8303\u7574\u5b66\uff0c\u4f46\u662f\u603b\u89c9\u5f97\u8303\u7574\u5b66\u7a7a\u6cdb\u7684\u597d\u50cf\u4ec0\u4e48\u90fd\u6ca1\u6709\u3002\u4ed6\u7684\u611f\u89c9\u6ca1\u9519\uff0c\u5f53\u7136\u4e5f\u4e0d\u53ea\u4ed6\u2f00\u4e2a\u2f08\u8fd9\u6837\u62b1\u6028\u3002\u4e8b\u5b9e\u4e0a\uff0c\u8303\u7574\u5b66\u662f\u548c\u96c6\u5408\u8bba\u2f00\u6837\u5e95\u5c42\u7684\u4e1c\u2ec4\u3002\u5c31\u8c61\u4f60\u53bb\u770b\u96c6\u5408\u8bba\u2f00\u6837\uff0c\u9664\u4e86\u2f00\u4e9b\u5f62\u5f0f\u7684\u5b9a\u4e49\uff0c\u4eff\u4f5b\u4ec0\u4e48\u90fd\u6ca1\u6709\u3002<\/span><\/p>\n


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\u5bf9\u7269\u7406\u5b66\u5bb6\u6765\u8bb2\uff0c\u770b\u96c6\u5408\u8bba\u2f0f\u4e4e\u6ca1\u6709\u4efb\u4f55\u2f64\u5904\uff0c\u771f\u6b63\u6709\u2f64\u7684\u662f\u5fae\u79ef\u5206\u548c\u7ebf\u6027\u4ee3\u6570\u3002\u6240\u4ee5\u4e5f\u53ea\u6709\u5f53\u4f60\u770b\u5230\u4e86\u8303\u7574\u5b66\u2f83\u2f30\u7684\u201c\u5fae\u79ef\u5206\u201d\u548c\u201c\u7ebf\u6027\u4ee3\u6570\u201d\u7684\u65f6\u5019\uff0c\u4f60\u624d\u80fd\u7406\u89e3\u5b83\u7684\u5f3a\u2f24\u3002\u6211\u8ba4\u4e3a\uff0cGrothendieck\u7684\u4ee3\u6570\u2f0f\u4f55\u5c31\u53ef\u4ee5\u7c97\u7565\u5730\u770b\u6210\u662f\u2f00\u4e2a\u65b0\u7684\u201c\u5fae\u79ef\u5206\u201d\uff0c\u2f7d\u5f20\u91cf\u8303\u7574\u7406\u8bba<\/span> (tensor category theory) <\/span>\u53ef\u4ee5\u770b\u6210\u662f\u2f00\u4e2a\u65b0\u7684\u201c\u7ebf\u6027\u4ee3\u6570\u201d\u3002\u8303\u7574\u5b66\u2fa5\u2faf\u7684\u201c\u5fae\u79ef\u5206\u201d<\/span>\uff08\u6216\u201c\u7ebf\u6027\u4ee3\u6570\u201d\uff09<\/span>\u90fd\u4e0d\u662f\u552f\u2f00\u7684\uff0c\u2f7d\u662f\u5343\u53d8\u4e07\u5316\u7684\u3002\u5bf9\u7269\u7406\u5b66\u6700\u6709\u2f64\u7684\u201c\u5fae\u79ef\u5206\u201d\u548c\u201c\u7ebf\u6027\u4ee3\u6570\u201d\u53ef\u80fd\u8fd8\u6ca1\u6709\u8bde\u2f63\u3002<\/span><\/p>\n


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\u4e0e\u96c6\u5408\u8bba\u4e0d\u540c\u7684\u662f\uff0c\u5bf9\u7269\u7406\u5b66\u5bb6\u6765\u8bf4\uff0c\u96c6\u5408\u8bba\u53ef\u4ee5\u5b8c\u5168\u5730\u5ffd\u7565\uff0c\u76f4\u63a5\u8df3\u5230\u5fae\u79ef\u5206\u548c\u7ebf\u6027\u4ee3\u6570\u4e0a\uff0c\u56e0\u4e3a\u96c6\u5408\u8bba\u7684\u8bed\u2f94\u548c\u57fa\u7840\u88ab\u51fd\u6570\u8bba\u7684\u8bed\u2f94\u8986\u76d6\u4e86\u3002\u4f46\u662f\u5bf9\u8303\u7574\u5b66\u6765\u8bb2\uff0c\u60f3\u8981\u8df3\u8fc7\u5979\u7684\u57fa\u7840\u8bed\u2f94\uff1a\u8303\u7574\u3001\u51fd\u2f26\u3001\u2f83\u7136\u53d8\u6362\u3001Yoneda\u5f15\u7406\uff0c\u76f4\u63a5\u5b66\u4e60\u5979\u7684\u201c\u5fae\u79ef\u5206\u201d\u548c\u201c\u7ebf\u6027\u4ee3\u6570\u201d\u662f\u4e0d\u53ef\u80fd\u7684\u3002\u9057\u61be\u7684\u662f\uff0c\u5230\u2f6c\u524d\u4e3a\u2f4c\uff0c\u8fd8\u6ca1\u6709\u2f00\u4e2a\u9002\u5408\u7269\u7406\u5b66\u5bb6\u8bfb\u7684\u8303\u7574\u5b66\u7684\u4e66\u3002<\/span><\/p>\n


<\/span><\/p>\n

\u8fd8\u6709\u2f00\u79cd\u8bef\u89e3\u662f\uff0c\u8303\u7574\u5b66\u5df2\u7ecf\u5efa\u2f74\u597d\u4e86\uff0c\u5b66\u597d\u4e86\u2f00\u672c\u8303\u7574\u5b66\u7684\u6570\u5b66\u4e66\uff0c\u5728\u7269\u7406\u4e0a\u7684\u5e94\u2f64\u53ef\u80fd\u5c31\u591f\u4e86\u3002\u5982\u679c\u4f60\u62b1\u7740\u8fd9\u6837\u7684\u2f3c\u6001\uff0c\u90a3\u4f60\u6ce8\u5b9a\u8981\u5931\u671b\u4e86\u3002\u2fb8\u5148\uff0c\u628a\u4efb\u4f55<\/span>\uff08\u4e0d\u8bba\u5979\u6709\u591a\u4e48\u4f18\u7f8e\uff09<\/span>\u6570\u5b66\u5957\u2f64\u5230\u7269\u7406\u4e0a\u7684\u60f3\u6cd5\u90fd\u662f\u7f18\u2f4a\u6c42\u2ee5\u7684\u505a\u6cd5\u3002\u53ea\u6709\u4ece\u7269\u7406\u5b9e\u9a8c\u6216\u7269\u7406\u56fe\u50cf\u51fa\u53d1\u2f7d\u53d1\u73b0\u7684\u6570\u5b66\u624d\u662f\u5bf9\u7269\u7406\u6709\u610f\u4e49\u7684\uff0c\u5982\u679c\u78b0\u5de7\u8fd9\u4e2a\u6570\u5b66\u5df2\u7ecf\u88ab\u6570\u5b66\u5bb6\u53d1\u73b0\uff0c\u90a3\u4e5f\u53ea\u662f\u5076\u7136\u60c5\u51b5\u2f7d\u5df2\u3002\u901a\u5411\u672a\u77e5\u4e4b\u2ed4\u2f24\u591a\u662f\u6ca1\u6709\u73b0\u6210\u7684\u94a5\u5319\u7684\u3002<\/span><\/p>\n


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\u7269\u7406\u5b66\u771f\u6b63\u9700\u8981\u7684\u8303\u7574\u5b66\u7edd\u2f24\u591a\u6570\u8fd8\u4e0d\u5b58\u5728\uff0c\u9700\u8981\u6211\u4eec\u53bb\u2f00\u8fb9\u53d1\u5c55\u7269\u7406\uff0c\u2f00\u8fb9\u53d1\u5c55\u6570\u5b66\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u2fa5\u2faf\uff0c\u65b0\u7684\u7269\u7406\u548c\u65b0\u7684\u6570\u5b66\u6ca1\u6709\u533a\u522b\uff0c\u5b83\u4eec\u90fd\u662f\u2f24\u2f83\u7136\u7684\u9690\u85cf\u7ed3\u6784\u3002\u73b0\u5728\u7684\u8303\u7574\u5b66\u8fd8\u5728\u53d1\u5c55\u7684\u521d\u7ea7\u9636\u6bb5\uff0c\u5fae\u79ef\u5206\u53ef\u4ee5\u53d1\u5c55\u2f0f\u767e\u5e74\u7684\u8bdd\uff0c\u8303\u7574\u5b66\u2f24\u6982\u4e5f\u9700\u8981\u2f0f\u767e\u5e74\u3002\u2f7d\u6211\u8fd9\u4e9b\u5e74\u7684\u5b9e\u8df5\u544a\u8bc9\u6211\uff0c\u7269\u7406\u80fd\u7ed9\u6211\u4eec\u5e26\u6765\u7684\u65b0\u7684\u3001\u8d85\u8d8a\u6570\u5b66\u5bb6\u60f3\u8c61\u2f12\u7684\u8303\u7574\u5b66\u624d\u662f\u771f\u7684\u6ce2\u6f9c\u58ee\u9614\u3002<\/span><\/p>\n


<\/p>\n

\n
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\u81f4\u8c22<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

\u611f\u8c22\u5fb7\u56fd\u54e5\u5ef7\u6839\u2f24\u5b66\u7684\u6731\u6668\u7545\u2f7c\u5e08\u3001\u6e05\u534e\u2f24\u5b66\u2fbc\u7b49\u7814\u7a76\u9662\u7684\u6c6a\u5fe0\u2f7c\u5e08\u548c\u4e18\u6210\u6850\u6570\u5b66\u4e2d\u2f3c\u7684\u2f65\u57a0\u2f7c\u5e08\u3001\u5357\u79d1\u2f24\u91cf\u2f26\u79d1\u5b66\u4e0e\u2f2f\u7a0b\u7814\u7a76\u9662\u7684\u5434\u548f\u65f6\u2f7c\u5e08\u548c\u90d1\u6d69\u2f7c\u5e08\u3001\u9ebb\u7701\u7406\u2f2f\u5b66\u9662\u7684\u2f42\u2f29\u521a\u2f7c\u5e08\u3001\u4e2d\u79d1\u9662\u7269\u7406\u6240\u7684\u66f9\u5219\u8d24\u2f7c\u5e08\u548c\u65af\u5766\u798f\u2f24\u5b66\u7684\u7941\u6653\u4eae\u2f7c\u5e08\u63d0\u51fa\u7684\u5f88\u591a\u5b9d\u8d35\u610f\u2ec5\u3002<\/span><\/p>\n


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\u6ce8\u91ca<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

[1] \u8fd9\u2fa5\u2faf\u6211\u4eec\u8fd8\u662f\u2f64\u5230\u4e86\u201c\u552f\u2f00\u201d\u8fd9\u4e2a\u6982\u5ff5\uff0c\u597d\u50cf\u662f\u5faa\u73af\u5b9a\u4e49\u201c1\u201d\u7684\u610f\u601d\u3002\u5176\u5b9e\u6211\u4eec\u53ef\u4ee5\u4ece\u6280\u672f\u4e0a\u56de\u907f\u5b83\uff0c\u2f50\u5982\uff1a\u6211\u4eec\u53ef\u4ee5\u8bf4\u4e07\u6709\u6027\u8d28\u4e2d\u7684\uff08\u8ba9\u56fe\u8868\u4ea4\u6362\u7684\uff09\u6620\u5c04\u6784\u6210\u7684\u96c6\u5408\u5b58\u5728\u5230\u96c6\u5408{O}\u7684\u53cc\u5c04\u3002<\/span><\/p>\n

\u6211\u4eec\u8fd9\u2fa5\u5e76\u4e0d\u662f\u60f3\u63a2\u8ba8\u6570\u5b66\u7684\u57fa\u7840\uff0c\u2f7d\u662f\u5c55\u73b0\u2f00\u79cd\u5bf91\u548c1+1\u7684\u5168\u65b0\u89e3\u8bfb\u3002\u4e0d\u8fc7\u4ece\u4e07\u6709\u6027\u8d28\u2fa5\u2faf\u4e0d\u65ad\u5730\u2f64\u201c\u5b58\u5728\u201d\u548c\u201c\u552f\u2f00\u201d\u53ef\u4ee5\u770b\u51fa\uff0c\u5728\u2f83\u7136\u54f2\u5b66\u7684\u610f\u4e49\u4e0b\uff0c\u201c\u552f\u2f00\u201d\u6709\u53ef\u80fd\u662f\u548c\u201c\u5b58\u5728\u201d\u540c\u7b49\u57fa\u672c\u7684\u6982\u5ff5\u3002<\/span><\/p>\n

[2] Boundary-bulk relation in topological orders, Liang Kong, Xiao-Gang Wen, Hao Zheng, Nucl. Phys. B 922 (2017), 62-76 [arXiv:1702.00673]<\/span><\/p>\n


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\u8303\u7574\u8bba\u7b2c\u4e8c\u5b63\uff1a\u8de8\u5b66\u79d1\u7684\u79d1\u5b66\u65b9\u6cd5\u8bba<\/strong><\/span><\/p>\n

\u6bcf\u5468\u66f4\u65b0\uff0c\u6301\u7eed\u62a5\u540d\u4e2d
<\/strong><\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


<\/section>\n

\u8bfe\u7a0b\u5927\u7eb2\uff08\u7b2c\u4e8c\u5b63\uff09<\/span><\/strong><\/span><\/h3>\n
    \n
  1. \n

    \u7528\u7bad\u5934\u6784\u9020\u77e9\u9635\uff08\u514d\u8d39\u516c\u5f00\uff09<\/p>\n<\/li>\n

  2. \n

    \u51fd\u5b50\u8303\u7574\uff08\u514d\u8d39\u516c\u5f00\uff09<\/p>\n<\/li>\n

  3. \n

    \u53ef\u8868\u51fd\u5b50I<\/p>\n<\/li>\n

  4. \n

    \u53ef\u8868\u51fd\u5b50II<\/p>\n<\/li>\n

  5. \n

    Yoneda\u5f15\u7406<\/p>\n<\/li>\n

  6. \n

    \u4f34\u968f\u51fd\u5b50<\/p>\n<\/li>\n

  7. \n

    \u5f20\u91cf\u79ef<\/p>\n<\/li>\n

  8. \n

    \u5f20\u91cf\u4ee3\u6570<\/p>\n<\/li>\n

  9. \n

    \u5e7a\u534a\u8303\u7574<\/p>\n<\/li>\n

  10. \n

    \u5355\u5b50<\/p>\n<\/li>\n

  11. \n

    \u6cdb\u6027\u8d28<\/p>\n<\/li>\n

  12. \n
    Abel\u8303\u7574<\/section>\n<\/li>\n<\/ol>\n


    <\/h3>\n


    <\/p>\n

    \u5b66\u4e60\u5efa\u8bae<\/span><\/strong><\/span><\/h3>\n


    <\/p>\n

    \u96c6\u667a\u8303\u7574\u8bba\u5165\u95e8\u7cfb\u5217\u8bfe\u7a0b\u7b2c\u4e00\u5b63\u7684\u5b66\u5458\u4ece\u5728\u8bfb\u7684\u5927\u5b66\u751f\u3001\u5728\u804c\u7684\u6570\u5b66\u7231\u597d\u8005\u3001\u7814\u7a76\u751f\u3001\u6559\u5b66\u79d1\u7814\u4eba\u58eb\uff0c\u5230\u4e00\u4e2a\u7814\u7a76\u8bfe\u9898\u7ec4\u7684\u5e08\u751f\u3002\u4ed6\u4eec\u7684\u7814\u7a76\u5174\u8da3\u5305\u62ec\u6570\u5b66\u3001\u7269\u7406\u3001\u91cf\u5b50\u4fe1\u606f\u3001\u4eba\u5de5\u667a\u80fd\u3001\u8ba1\u7b97\u7406\u8bba\u3001\u751f\u547d\u79d1\u5b66\u3001\u7ecf\u6d4e\u5b66\u7b49\uff0c\u8fd9\u65e2\u4f53\u73b0\u4e86\u96c6\u667a\u793e\u533a\u6d3b\u8dc3\u7684\u4ea4\u53c9\u5b66\u79d1\u5b66\u4e60\u6c1b\u56f4\uff0c\u4e5f\u4f53\u73b0\u4e86\u8303\u7574\u8bba\u7b2c\u4e00\u5b63\u7684\u8bbe\u8ba1\u521d\u8877\uff0c\u6253\u9020\u4e00\u4e2a\u201c\u4eba\u4eba\u53ef\u5b66\u201d\u7684\u8303\u7574\u8bba\u5165\u95e8\u8bfe\u7a0b\u3002\u7b2c\u4e00\u5b63\u6536\u5230\u4e86\u5f88\u591a\u5b66\u5458\u79ef\u6781\u7684\u53cd\u9988\uff0c\u60f3\u542c\u542c\u5927\u5bb6\u600e\u4e48\u8bf4\uff0c\u53ef\u89c1\u6587\u7ae0\uff1a<\/span>\u8303\u7574\u8bba\u4eba\u4eba\u7686\u53ef\u5165\u95e8\uff1f\u8001\u5e08\u548c\u540c\u5b66\u4eec\u8fd9\u6837\u8bf4\u2014\u2014<\/a><\/section>\n

    <\/section>\n
    \u5982\u679c\u60a8\u6ee1\u8db3\u4ee5\u4e0b\u4efb\u610f\u6761\u4ef6\uff0c\u4f46\u5c1a\u672a\u6709\u57fa\u7840\u7684\u8303\u7574\u8bba\u77e5\u8bc6\uff0c\u8bf7\u5148\u5b66\u4e60\u7b2c\u4e00\u5b63\u8bfe\u7a0b\uff1a<\/span><\/section>\n

    <\/section>\n