{"id":1329,"date":"2018-03-14T01:57:34","date_gmt":"2018-03-14T00:57:34","guid":{"rendered":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/?p=1329"},"modified":"2018-03-14T01:59:42","modified_gmt":"2018-03-14T00:59:42","slug":"general-affine-adjunctions-nullstellensatze-and-dualities","status":"publish","type":"post","link":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/1329-general-affine-adjunctions-nullstellensatze-and-dualities\/","title":{"rendered":"General affine adjunctions, Nullstellens\u00e4tze, and dualities"},"content":{"rendered":"<p>At last, we have finished and submitted our paper on &#8220;<a href=\"https:\/\/arxiv.org\/abs\/1412.8692\">General affine adjunctions, Nullstellens\u00e4tze, and dualities<\/a>&#8221; co-authored with <a href=\"http:\/\/www.oliviacaramello.com\">Olivia Caramello<\/a> and <a href=\"http:\/\/marra.di.unimi.it\">Vincenzo Marra<\/a>.<\/p>\n<blockquote>\n<div class=\"page\" title=\"Page 1\">\n<div class=\"section\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><strong>Abstract.<\/strong> We introduce and investigate a category-theoretic abstraction of the standard \u201csystem-solution\u201d adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert\u2019s <em>Nullstellensatz<\/em>, thereby achieving a complete characterisation of the fixed points on the algebraic side of the adjunction.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/blockquote>\n<p>The preprint is <a href=\"https:\/\/arxiv.org\/abs\/1412.8692\">available on arXiv<\/a>. \u00a0We made another preprint available some years ago(!), but the manuscript has changed in many respects. \u00a0The main differences between the two versions on arXiv are the following:<\/p>\n<ol>\n<li>The comparison with the existing literature is now more thorough.<\/li>\n<li>The categories R and D are now taken directly without passing through the quotient categories. In our opinion, this is cleaner and, as a consequence, it is now clearer what are the minimal assumption on the triplet I: T -&gt; S.<\/li>\n<li>There is now a section studying the issue of concreteness of the adjunction and comparing with the theory of concrete adjunction.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>At last, we have finished and submitted our paper on &#8220;General affine adjunctions, Nullstellens\u00e4tze, and dualities&#8221; co-authored with Olivia Caramello and Vincenzo Marra. Abstract. We introduce and investigate a category-theoretic abstraction of the standard \u201csystem-solution\u201d adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17],"tags":[30,82,179,180,55,89,84,81,79,181],"class_list":["post-1329","post","type-post","status-publish","format-standard","hentry","category-preprint","tag-adjunction","tag-algebraic-geometry","tag-category","tag-concrete-dualities","tag-duality","tag-gelfand-transform","tag-hilbert-nullstellensatz","tag-pontryagin-duality","tag-stone-duality","tag-syntax-and-semantics"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/1329","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/comments?post=1329"}],"version-history":[{"count":2,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/1329\/revisions"}],"predecessor-version":[{"id":1332,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/1329\/revisions\/1332"}],"wp:attachment":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/media?parent=1329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/categories?post=1329"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/tags?post=1329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}