{"id":1470,"date":"2022-02-12T19:59:29","date_gmt":"2022-02-12T11:59:29","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1470"},"modified":"2025-02-26T11:04:55","modified_gmt":"2025-02-26T03:04:55","slug":"generator_related_terminology_description","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/02\/12\/generator_related_terminology_description\/","title":{"rendered":"\u751f\u6210\u5143\u76f8\u5173\u672f\u8bed\u8bf4\u660e"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u6700\u8fd1\u5b66\u4e60\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u4ee3\u6570\u62d3\u6251\u8bfe\u7a0b\u65f6\u65f6\u5e38\u4f1a\u51fa\u73b0\u4e66\u4e0a\u672a\u4ecb\u7ecd\u8fc7\u7684\u672f\u8bed, \u6545\u4ee5\u6b64\u6587\u4f5c\u4e3a\u8bb0\u5f55.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/en.wikipedia.org\/wiki\/Universal_property\">Universal property<\/a><br \/>\n2. <a href=\"https:\/\/honghao0304.github.io\/docs\/grouptheory\/exact_sequence.pdf\">exact_sequence.pdf<\/a><\/p>\n<p><strong>1. \u5178\u5219\u5d4c\u5165<\/strong> <\/p>\n<p>\u5373\u5355\u5c04.<\/p>\n<p><strong>2. Universal Mapping Property<\/strong> <\/p>\n<p>\u8bbe$F: \\mathcal{C} \\to \\mathcal{D}$\u4e3a\u8303\u7574$\\mathcal{C}, \\mathcal{D}$\u4e4b\u95f4\u7684\u51fd\u5b50, $X$\u4e3a$\\mathcal{D}$\u4e2d\u7684\u4e00\u4e2a\u5bf9\u8c61, $A, A&#8217;$\u5747\u4e3a$\\mathcal{C}$\u4e2d\u7684\u5bf9\u8c61, \u5219\u51fd\u5b50$F$\u5206\u522b\u5c06$A, A&#8217;$\u4e0e$h \\in \\mathcal{C}$\u6620\u5c04\u5230$\\mathcal{D}$\u4e2d\u7684$F(A)$, $F(A&#8217;)$\u4e0e$F(h)$.<br \/>\n$\\\\$ \u4e00\u4e2a\u4ece$X$\u6620\u5c04\u5230$F$\u7684Universal Morphism\u662f$\\mathcal{D}$\u4e2d\u552f\u4e00\u7684\u5bf9$(A, u:<br \/>\n $$ X \\to F( $$ A))$, \u8be5\u5bf9\u5177\u6709\u5982\u4e0b\u6027\u8d28, \u901a\u5e38\u79f0\u4e4b\u4e3aUniversal Mapping Property: \u5bf9\u4e8e$\\mathcal{D}$\u4e2d\u4efb\u610f\u5f62\u5f0f\u4e3a$f: X \\to F(A&#8217;)$\u7684\u6001\u5c04, \u5b58\u5728$\\mathcal{C}$\u4e2d\u552f\u4e00\u7684\u6001\u5c04$h: A \\to A&#8217;$, \u4f7f\u5f97\u5982\u4e0b\u4ea4\u6362\u56fe\u8868\u6210\u7acb.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Universal-Mapping-Property1.png\" alt=\"\" width=\"375\" height=\"185\" class=\"aligncenter size-full wp-image-1477\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Universal-Mapping-Property1.png 375w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Universal-Mapping-Property1-300x148.png 300w\" sizes=\"(max-width: 375px) 100vw, 375px\" \/><\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u6982\u5ff5\u5bf9\u5076\u5316. \u4e00\u4e2a\u4ece$F$\u6620\u5c04\u5230$X$\u7684Universal Morphism\u662f$\\mathcal{D}$\u4e2d\u552f\u4e00\u7684\u5bf9$(A, u: F(A) \\to X)$, \u8be5\u5bf9\u5177\u6709\u5982\u4e0b\u6027\u8d28, \u901a\u5e38\u79f0\u4e4b\u4e3aUniversal Mapping Property: \u5bf9\u4e8e$\\mathcal{D}$\u4e2d\u4efb\u610f\u5f62\u5f0f\u4e3a$f: F(A&#8217;) \\to X$\u7684\u6001\u5c04, \u5b58\u5728$\\mathcal{C}$\u4e2d\u552f\u4e00\u7684\u6001\u5c04$h: A&#8217; \\to A$, \u4f7f\u5f97\u5982\u4e0b\u4ea4\u6362\u56fe\u8868\u6210\u7acb.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Universal-Mapping-Property2.png\" alt=\"\" width=\"374\" height=\"192\" class=\"aligncenter size-full wp-image-1482\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Universal-Mapping-Property2.png 374w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Universal-Mapping-Property2-300x154.png 300w\" sizes=\"(max-width: 374px) 100vw, 374px\" \/><\/p>\n<p><strong>3. Fibered Coproduct\u4e0eCocartesian\u56fe\u8868<\/strong> <\/p>\n<p>\u8bbe$\\mathcal{C}$\u4e3a\u4e00\u4e2a\u8303\u7574, \u8bbe$f: Z \\to X$, $g: Z \\to Y$, $f$\u4e0e$g$\u7684Fibered Coproduct\u662f\u6307\u6570\u636e$(W, p_1, p_2)$, \u5176\u4e2d$W \\in \\mathcal{O}b(\\mathcal{C})$, $p_1: X \\to $$ W$, $p_2: Y \\to W$, \u4e14$(W, p_1, $$ p_2)$\u6ee1\u8db3\u5982\u4e0b\u7ea6\u675f: \u5bf9\u4e8e\u4efb\u610f\u4f7f\u5f97\u4e0b\u9762\u56fe\u8868\u53ef\u4ea4\u6362\u7684\u4e24\u4e2a\u6620\u5c04$u: X \\to U$, $v: Y $$ \\to U$,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Fibered-Coproduct.png\" alt=\"\" width=\"383\" height=\"317\" class=\"aligncenter size-full wp-image-1491\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Fibered-Coproduct.png 1148w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Fibered-Coproduct-300x248.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Fibered-Coproduct-768x636.png 768w\" sizes=\"(max-width: 383px) 100vw, 383px\" \/><\/p>\n<p>\u5b58\u5728\u4e00\u4e2a\u552f\u4e00\u7684\u6001\u5c04$h: W \\to U$, s.t. $h \\circ p_1 = u$, $h \\circ p_2 = v$. \u6b64\u65f6, \u79f0\u5982\u4e0b\u56fe\u8868\u4e3a\u4e00\u4e2aCocartesian\u56fe\u8868.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian.png\" alt=\"\" width=\"221\" height=\"191\" class=\"aligncenter size-full wp-image-1499\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian.png 663w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian-300x260.png 300w\" sizes=\"(max-width: 221px) 100vw, 221px\" \/><\/p>\n<p><strong>Rmk1.<\/strong> \u5c06\u4ee5\u4e0a\u8bba\u8ff0\u4e2d\u7684\u7bad\u5934\u5168\u90e8\u53cd\u5411\u540e, \u5373\u53ef\u5f97Fibered Product\u7684\u5b9a\u4e49.<br \/>\n$\\\\$ <strong>Rmk2.<\/strong> \u5728\u8303\u7574$\\mathcal{C}$\u4e2d$f$\u4e0e$g$\u7684Fibered Coproduct\u5728\u5dee\u4e00\u4e2a\u540c\u6784\u7684\u610f\u4e49\u4e0b\u662f\u552f\u4e00\u7684.<\/p>\n<p><strong>4. \u81ea\u7531\u7fa4<\/strong><\/p>\n<p>\u5728\u4e4b\u524d\u7684\u6587\u7ae0\u4e2d\u5176\u5b9e\u5df2\u7ecf\u4ecb\u7ecd\u8fc7\u81ea\u7531\u7fa4\u7684\u5b9a\u4e49\u4e86, \u6b64\u5904\u518d\u4ecb\u7ecd\u4e00\u4e0b\u5728\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u4ee3\u6570\u62d3\u6251\u8bfe\u7a0b\u4e2d\u63d0\u53ca\u7684\u81ea\u7531\u7fa4\u7684\u5b9a\u4e49, \u8fd9\u91cc\u662f\u5229\u7528\u81ea\u7531\u79ef\u6765\u5b9a\u4e49\u81ea\u7531\u7fa4\u7684.<br \/>\n$\\\\$ \u8bbe$S$\u4e3a\u4e00\u4e2a\u96c6\u5408, $S$\u4e0a\u7684\u81ea\u7531\u7fa4$F(S)$\u5b9a\u4e49\u4e3a:$$F(S) := *_{\\alpha \\in S} \\left \\langle \\alpha \\right \\rangle,$$\u5176\u4e2d$\\left \\langle \\alpha \\right \\rangle = \\{ \\alpha^n \\ | \\ n \\in Z \\}$\u662f\u4e00\u4e2a\u65e0\u9650\u5faa\u73af\u7fa4.<\/p>\n<p><strong>5. \u77ed\u6b63\u5408\u5e8f\u5217<\/strong><\/p>\n<p><strong>5.1 \u6b63\u89c4\u5b50\u7fa4\u4e0e\u5546\u7fa4<\/strong><\/p>\n<p>\u9996\u5148\u6211\u4eec\u56de\u987e\u6b63\u89c4\u5b50\u7fa4\u7684\u5b9a\u4e49. \u8bbe$H$\u662f$G$\u7684\u5b50\u7fa4, \u5219\u82e5\u5bf9\u4e8e$\\forall g \\in G$\u90fd\u6709$gHg^{-1} $$ = H$, \u5219$H$\u79f0\u4e3a$G$\u7684\u6b63\u89c4\u5b50\u7fa4. \u6b63\u89c4\u5b50\u7fa4\u7684\u5b9a\u4e49\u7acb\u523b\u5bfc\u81f4\u4e24\u4e2a\u63a8\u8bba, \u5373<br \/>\n$\\\\$ a) \u5de6\u53f3\u966a\u96c6\u76f8\u540c, $gH = Hg$, \u56e0\u6b64\u5de6\u53f3\u966a\u96c6\u7a7a\u95f4\u662f\u4e00\u6837\u7684, \u5e76\u5b9a\u4e49\u4e3a\u5546\u7a7a\u95f4$G \/ $$ H$.<br \/>\n$\\\\$ b) \u76f8\u5e94\u7684\u5546\u7a7a\u95f4$G \/ H$\u53ef\u4ee5\u4ece$G$\u7ee7\u627f\u7fa4\u4e58\u6cd5\u7ed3\u6784, \u5e76\u6784\u6210\u5546\u7fa4.<br \/>\n$\\\\$ \u5176\u4e2d, \u6211\u4eec\u6ce8\u610f\u5230\u4e24\u4e2a\u8fd1\u4e4e\u5e73\u51e1\u7684\u8bba\u65ad, \u7b2c\u4e00\u662f&#8221;$H$\u662f$G$\u7684\u5b50\u96c6&#8221;, \u7b2c\u4e8c\u662f&#8221;$G \/ H$\u7684\u5143\u7d20$[gH]$\u90fd\u5bf9\u5e94\u4e00\u4e2a$G$\u4e2d$H$\u7684\u966a\u96c6$gH$&#8221;. \u6211\u4eec\u53ef\u4ee5\u628a\u8fd9\u4e24\u4e2a\u8bba\u65ad\u8868\u8fbe\u4e3a, \u5b58\u5728\u4e00\u4e2a\u5355\u5c04(Injection), \u79f0\u4e3a&#8221;\u6620\u5165\u6620\u5c04&#8221;(Inclusion Map) $\\imath : H \\to G$, \u4ee5\u53ca\u6ee1\u5c04(Surjection), \u79f0\u4e3a&#8221;\u6295\u5f71\u6620\u5c04&#8221;(Projection) $\\pi : G \\to G \/ H$, \u4f7f\u5f97$\\pi(gH) = $$ [gH]$(\u5176\u4e2d$\\imath$\u548c$\\pi$\u5206\u522b\u662f&#8221;Inclusion&#8221; \u548c&#8221;Projection&#8221; \u7684\u5e0c\u814a\u9996\u5b57\u6bcd). \u90a3\u4e48\u6839\u636e\u540c\u6001\u6838\u4e0e$G \/ H$\u7684\u5b9a\u4e49, \u6211\u4eec\u6709$$Im \\imath = ker \\pi,$$\u56e0\u4e3a$Im \\imath = H \\subset G$\u6b63\u662f\u88ab\u89c6\u4e3a$G \/ H$\u4e2d\u7684\u5355\u4f4d\u5143, \u6216\u8005\u7b49\u4ef7\u5730\u8bf4, \u6574\u4e2a$Im \\imath$\u88ab$\\pi$\u6620\u5c04\u4e3a\u5355\u4f4d\u5143.<\/p>\n<p><strong>5.2 \u77ed\u6b63\u5408\u5e8f\u5217<\/strong><\/p>\n<p>\u4e0a\u9762\u8fd9\u4e9b\u8bba\u65ad\u4e0e\u5173\u7cfb\u5f0f\u53ef\u4ee5\u88ab\u91cd\u65b0\u7ec4\u5408\u4e3a&#8221;\u77ed\u6b63\u5408\u5e8f\u5217&#8221;(Short Exact Sequence, \u6709\u65f6&#8221;\u6b63\u5408&#8221; \u4e5f\u7ffb\u8bd1\u4e3a&#8221;\u6070\u5f53&#8221;) \u7684\u6982\u5ff5.<br \/>\n$\\\\$ \u8003\u8651\u56db\u4e2a\u9996\u5c3e\u76f8\u63a5\u7684\u540c\u6001\u6620\u5c04,$$\\{ e \\} \\overset{\\varphi_0}{\\longrightarrow} G_1 \\overset{\\varphi_1}{\\longrightarrow} G_2 \\overset{\\varphi_2}{\\longrightarrow} G_3 \\overset{\\varphi_3}{\\longrightarrow} \\{ e \\},$$\u8be5\u5e8f\u5217\u662f&#8221;\u6b63\u5408\u7684&#8221; \u8bf4\u7684\u662f\u8be5\u5e8f\u5217\u6bcf\u4e24\u4e2a\u76f8\u90bb\u6620\u5c04\u90fd\u6ee1\u8db3$$Im \\varphi_i = ker \\varphi_{i + 1}.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u5b66\u4e60\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u4ee3\u6570\u62d3\u6251\u8bfe\u7a0b\u65f6\u65f6\u5e38\u4f1a\u51fa\u73b0\u4e66\u4e0a\u672a\u4ecb\u7ecd\u8fc7\u7684\u672f\u8bed, \u6545\u4ee5\u6b64\u6587\u4f5c\u4e3a\u8bb0\u5f55.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1470"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1470"}],"version-history":[{"count":43,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1470\/revisions"}],"predecessor-version":[{"id":3613,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1470\/revisions\/3613"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1470"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1470"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1470"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}