{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T16:35:09Z","timestamp":1772037309299,"version":"3.50.1"},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11607,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,6]]},"abstract":"<jats:p>Let <jats:italic>T<\/jats:italic> be a countable complete theory and <jats:italic>C<\/jats:italic>(<jats:italic>T<\/jats:italic>) the category whose objects are the models of <jats:italic>T<\/jats:italic> and morphisms are the elementary maps. The main object of this paper will be the study of <jats:italic>C<\/jats:italic>(<jats:italic>T<\/jats:italic>). The idea that a better understanding of the category may give us model theoretic information about <jats:italic>T<\/jats:italic> is quite natural: The (semi) group of automorphisms (endomorphisms) of a given structure is often a powerful tool for studying this structure. But certainly, one of the very first questions to be answered is: \u201cto what extent does this category <jats:italic>C<\/jats:italic>(<jats:italic>T<\/jats:italic>) determine <jats:italic>T<\/jats:italic>?\u201d<\/jats:p><jats:p>There is some obvious limitation: for example let <jats:italic>T<\/jats:italic><jats:sub>0<\/jats:sub> be the theory of infinite sets (in a language containing only =) and <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub> the theory, in the language ( =, <jats:italic>U<\/jats:italic>(<jats:italic>\u03bd<\/jats:italic><jats:sub>0<\/jats:sub>),<jats:italic>f<\/jats:italic>(<jats:italic>\u03bd<\/jats:italic><jats:sub>0<\/jats:sub>)) stating that:<\/jats:p><jats:p>(1) <jats:italic>U<\/jats:italic> is infinite.<\/jats:p><jats:p>(2)<jats:italic>f<\/jats:italic> is a bijective map from <jats:italic>U<\/jats:italic> onto its complement.<\/jats:p><jats:p>It is quite easy to see that <jats:italic>C<\/jats:italic>(<jats:italic>T<\/jats:italic><jats:sub>0<\/jats:sub>) is equivalent to <jats:italic>C<\/jats:italic>(<jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub>). But, in this case, <jats:italic>T<\/jats:italic><jats:sub>0<\/jats:sub> and  <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub> can be \u201cinterpreted\u201d each in the other. To make this notion of interpretation precise, we shall associate with each theory <jats:italic>T<\/jats:italic> a category, loosely denoted by <jats:italic>T<\/jats:italic>, defined as follows:<\/jats:p><jats:p>(1) The objects are the formulas <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044224_inline01\"\/> in the given language.<\/jats:p><jats:p>(2) The morphisms from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044224_inline01\"\/> into <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044224_inline01\"\/> are the formulas <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044224_inline02\"\/> such that<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200044224_eqnU01\"\/><\/jats:disp-formula><\/jats:p><jats:p>(i.e. <jats:italic>f<\/jats:italic> defines a map from <jats:italic>\u03d5<\/jats:italic> into <jats:italic>\u03d5<\/jats:italic>; two morphisms defining the same map in all models of <jats:italic>T<\/jats:italic> should be identified).<\/jats:p>","DOI":"10.2307\/2273140","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:59:42Z","timestamp":1146952782000},"page":"249-266","source":"Crossref","is-referenced-by-count":30,"title":["On the category of models of a complete theory"],"prefix":"10.1017","volume":"47","author":[{"given":"Daniel","family":"Lascar","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044224_ref005","volume-title":"Classification theory and the number of non-isomorphic models, Studies in logic","author":"Shelah","year":"1978"},{"key":"S0022481200044224_ref001","first-page":"330","volume":"44","author":"Lascar","year":"1979","journal-title":"An introduction to forking"},{"key":"S0022481200044224_ref004","volume-title":"Lecture Notes in Mathematics","volume":"611","author":"Makkai"},{"key":"S0022481200044224_ref003","unstructured":"Makkai M. , The topos of types (to appear)."},{"key":"S0022481200044224_ref002","unstructured":"Makkai M. , Full continuous embeddings of toposes (to appear)."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044224","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T21:15:58Z","timestamp":1558732558000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044224\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,6]]},"references-count":5,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1982,6]]}},"alternative-id":["S0022481200044224"],"URL":"https:\/\/doi.org\/10.2307\/2273140","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,6]]}}}