{"id":10,"date":"2023-05-14T17:58:27","date_gmt":"2023-05-14T17:58:27","guid":{"rendered":"http:\/\/aprilg.moe\/?p=10"},"modified":"2025-12-07T23:27:36","modified_gmt":"2025-12-07T15:27:36","slug":"ex-the-rising-sea-2-7-f","status":"publish","type":"post","link":"https:\/\/aprilg.moe\/?p=10","title":{"rendered":"Ex: The Rising Sea, 2.7.F"},"content":{"rendered":"\n<p>Warning after two years:<\/p>\n\n\n\n<p>Please take this proof with a grain of salt. This proof is long and has lots of details, so expect lots of typos. Two years have passed, and things didn&#8217;t go well. I still believe the idea is correct (but inefficient) though.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>I had difficulties with this problem, and found someone already posted a <a href=\"https:\/\/math.stackexchange.com\/questions\/4609839\/canonicity-of-the-push-pull-map-for-sheaves-of-sets\" data-type=\"URL\" data-id=\"https:\/\/math.stackexchange.com\/questions\/4609839\/canonicity-of-the-push-pull-map-for-sheaves-of-sets\">question<\/a> on MSE with no categorical answer present. (UPDATE: Now there is a good one thanks to Yizhen Chen.) After a week of frustration, I came up with this fully categorical approach:<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/aprilg.moe\/wp-content\/uploads\/2025\/12\/pushpull.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of pushpull.pdf.\"><\/object><a id=\"wp-block-file--media-b1bc7c3b-692e-48b9-a321-380279ac1590\" href=\"https:\/\/aprilg.moe\/wp-content\/uploads\/2025\/12\/pushpull.pdf\">pushpull.pdf<\/a><a href=\"https:\/\/aprilg.moe\/wp-content\/uploads\/2025\/12\/pushpull.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-b1bc7c3b-692e-48b9-a321-380279ac1590\">Download<\/a><\/div>\n\n\n\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-0324f3f6-c738-4bcb-8e3f-86a154b74d6e\" href=\"https:\/\/aprilg.moe\/wp-content\/uploads\/2025\/12\/pushpull.tex\">pushpull.tex<\/a><a href=\"https:\/\/aprilg.moe\/wp-content\/uploads\/2025\/12\/pushpull.tex\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-0324f3f6-c738-4bcb-8e3f-86a154b74d6e\">Download<\/a><\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Warning after two years: Please take this proof with a grain of salt. This proof is long and has lots of details, so expect lots of typos. Two years have passed, and things didn&#8217;t go well. I still believe the idea is correct (but inefficient) though. I had difficulties with this problem, and found someone [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-10","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/aprilg.moe\/index.php?rest_route=\/wp\/v2\/posts\/10","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/aprilg.moe\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/aprilg.moe\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/aprilg.moe\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/aprilg.moe\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10"}],"version-history":[{"count":3,"href":"https:\/\/aprilg.moe\/index.php?rest_route=\/wp\/v2\/posts\/10\/revisions"}],"predecessor-version":[{"id":16,"href":"https:\/\/aprilg.moe\/index.php?rest_route=\/wp\/v2\/posts\/10\/revisions\/16"}],"wp:attachment":[{"href":"https:\/\/aprilg.moe\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/aprilg.moe\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/aprilg.moe\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}