{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:41:08Z","timestamp":1753890068717,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2017,11,30]],"date-time":"2017-11-30T00:00:00Z","timestamp":1512000000000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2017,11,30]],"date-time":"2017-11-30T00:00:00Z","timestamp":1512000000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2017,11,30]],"date-time":"2017-11-30T00:00:00Z","timestamp":1512000000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>We show that a version of Martin-L\\&amp;quot;of type theory with an extensional identity type former I, a unit type N1 , Sigma-types, Pi-types, and a base type is a free category with families (supporting these type formers) both in a 1- and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We show that equality in this category is undecidable by reducing it to the undecidability of convertibility in combinatory logic. Essentially the same construction also shows a slightly strengthened form of the result that equality in extensional Martin-L\\&amp;quot;of type theory with one universe is undecidable.<\/jats:p>","DOI":"10.23638\/lmcs-13(4:22)2017","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:35:24Z","timestamp":1743701724000},"source":"Crossref","is-referenced-by-count":2,"title":["Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended version)"],"prefix":"10.23638","volume":"Volume 13, Issue 4","author":[{"given":"Simon","family":"Castellan","sequence":"first","affiliation":[]},{"given":"Pierre","family":"Clairambault","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Dybjer","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2017,11,30]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1504.03995v3","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1504.03995v3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:35:24Z","timestamp":1743701724000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/4113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,30]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-13(4:22)2017","relation":{"is-same-as":[{"id-type":"arxiv","id":"1504.03995","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1504.03995","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2017,11,30]]},"article-number":"4113"}}