{"id":605,"date":"2014-12-29T11:19:57","date_gmt":"2014-12-29T10:19:57","guid":{"rendered":"http:\/\/logica.dmi.unisa.it\/lucaspada\/?p=605"},"modified":"2017-10-05T09:41:56","modified_gmt":"2017-10-05T08:41:56","slug":"another-duality-for-the-whole-variety-of-mv-algebras","status":"publish","type":"post","link":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/605-another-duality-for-the-whole-variety-of-mv-algebras\/","title":{"rendered":"A(nother) duality for the whole variety of MV-algebras"},"content":{"rendered":"

This is the abstract of a talk I gave in Florence at Beyond 2014<\/a>.<\/p>\n

Given a category C<\/span> one can form its ind-completion<\/i> by taking all formal directed colimits of objects in C<\/span>. The “correct” arrows to consider are then families of some special equivalence classes of arrows in C<\/span> (Johnstone 1986, V.1.2, pag. 225). The pro-completion<\/i> is formed dually by taking all formal directed limits. For general reasons, the ind-completion of a category C<\/span> is dually equivalent to the pro<\/i>-completion of the dual category C^{\\rm op}<\/span>.<\/p>\n

$$\\textrm{ind}\\mbox{-}C\\simeq (\\textrm{pro}\\mbox{-}(C^{\\rm{op}}))^{\\rm{op}}. \u00a0 \u00a0 \u00a0 \\qquad\\qquad\u00a0(1)$$<\/p>\n

Ind- and pro- completions are very useful objects (as they are closed under directed (co)limits) but cumbersome to use, because of the involved definitions of arrows between objects. We prove that if C<\/span> is an algebraic category, then the situation considerably simplifies.<\/p>\n

If V<\/span> is any variety of algebras, one can think of any algebra A<\/span> in V<\/span> as colimit of finitely presented algebras as follows.<\/p>\n

Consider a presentation of A<\/span> i.e., a cardinal \\mu<\/span> and a congruence [\/latex]\\theta[\/latex] on the free \\mu<\/span>-generated algebra \\mathcal{F}(\\mu)<\/span> such that A\\cong \\mathcal{F}(\\mu)\/\\theta<\/span>. Now, consider the set F(\\theta)<\/span> of all finitely generated congruences contained in \\theta<\/span>, this gives a directed diagram in which the objects are the finitely presented algebras of the form \\mathcal{F}(n)\/\\theta_{i}<\/span> where \\theta_{i}\\in F(\\theta)<\/span> and X_{1},...,X_{n}<\/span> are the free generators occurring in \\theta_{i}<\/span>. It is straightforward to see that this diagram is directed, for if \\mathcal{F}(m)\/\\theta_{1}<\/span> and \\mathcal{F}(n)\/\\theta_{2}<\/span> are in the diagram, then both map into \\mathcal{F}(m+n)\/\\langle\\theta_{1}\\uplus\\theta_{2}\\rangle<\/span>, where \\langle\\theta_{1}\\uplus\\theta_{2}\\rangle<\/span> is the congruence generated by the disjoint union of \\theta_{1}<\/span> and \\theta_{2}<\/span>. Now, the colimit of such a diagram is exactly A<\/span>.<\/p>\n

Denoting by V_{\\textrm{fp}}<\/span> the full subcategory of V<\/span> of finitely presented objects, the above reasoning entails<\/p>\n

$$V\\simeq\\textrm{ind}\\mbox{-}V_{\\textrm{fp}}. \u00a0 \u00a0 \u00a0 \u00a0\\qquad\\qquad\u00a0(2)$$<\/p>\n

We apply our result to the special case where V<\/span> is the class of MV-algebras. One can then combine the duality between finitely presented MV-algebras and the category P_{\\mathbb{Z}}<\/span> of rational polyhedra with \\mathbb{Z}<\/span>-maps\u00a0(see here<\/a>), with (1)\u00a0\u00a0and (2)\u00a0to obtain,<\/p>\n

$$MV\\simeq\\textrm{ind}\\mbox{-}MV_{\\textrm{fp}}\\simeq \\textrm{pro}\\mbox{-}(P_{\\mathbb{Z}})^{\\rm{op}}. \u00a0\\qquad\\qquad\u00a0(3)$$<\/p>\n

This gives a categorical duality for the whole class of MV-algebras whose geometric content may be more transparent than other dualities in literature. In increasing order of complexity one has that any MV-algebra:<\/p>\n

    \n
  1. is dual to a polyhedron (Finitely presented case);<\/li>\n
  2. is dual to an intersection of polyhedra (Semisimple case);<\/li>\n
  3. is dual to a countable nested sequence of polyhedra (Finitely generated case);<\/li>\n
  4. is dual to the directed limit of a family of polyhedra. (General case).<\/li>\n<\/ol>\n<\/blockquote>\n

    Here<\/a>\u00a0are the slides of this talk<\/p>\n","protected":false},"excerpt":{"rendered":"

    This is the abstract of a talk I gave in Florence at Beyond 2014. Given a category one can form its ind-completion by taking all formal directed colimits of objects in . The “correct” arrows to consider are then families of some special equivalence classes of arrows in (Johnstone 1986, V.1.2, pag. 225). The pro-completion […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[42,53],"tags":[31,9,19,55,121,28,29,38,122,37],"class_list":["post-605","post","type-post","status-publish","format-standard","hentry","category-conferences","category-talk","tag-categorical-duality","tag-conference","tag-directed-colimit","tag-duality","tag-ind-completion","tag-lukasiewicz-logic","tag-mv-algebras","tag-piecewise-linear-maps","tag-pro-completion","tag-rational-polyhedra"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/605","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=605"}],"version-history":[{"count":15,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/605\/revisions"}],"predecessor-version":[{"id":1140,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/605\/revisions\/1140"}],"wp:attachment":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/media?parent=605"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/categories?post=605"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/tags?post=605"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}