[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82658-en":3,"doc-seo-82658-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82658,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Conceptual Completeness for Subgeometric Logics","Explores conceptual completeness for a fragment of geometric logic within the framework of the first and third authors. Reframes the traditional syntax-from-semantics view by characterizing conceptual completeness through a duality between theories and topoi. Proves that conceptually complete fragments embed conservatively into full geometric logic, offers new proof for coherent logic, and establishes conceptual completeness for regular, disjunctive, and essentially algebraic logic with falsum. Shows equivalence to a reconstruction result under set-based completeness, recovering Makkai’s theorem via ultracategories.","arXiv :2607 .02250v1 [math .LO] 2 Jul 2026  \nCONCEPTUAL COMPLETENESS FOR SUBGEOMETRIC LOGICS  \nIVAN DI LIBERTI, UMBERTO TARANTINO, AND LINGYUAN YE  \nAbstract . We explore the notion of conceptual completeness for a fragment of geometric logic in the framework developed by the first and third author.  \nUnlike its traditional interpretation as a reconstruction of syntax from semantics, in this paper we characterise conceptual completeness of a fixed fragment in terms of a duality between theories and topoi. We then show that conceptually complete fragments are conservatively embedded in full geometric logic, thus casting conceptual completeness in a new proof-theoretic light. We give a new proof of conceptual completeness for coherent logic, and we also show that regular, disjunctive, and essentially algebraic logic with falsum are conceptually complete. Finally, we show that our notion is equivalent to a traditional reconstruction result under the assumption of completeness with respect to set-based models: in the coherent case, we thus recover Makkai’s original reconstruction theorem via ultracategories.  \nKeywords. fragment of geometric logic, conceptual completeness, categorical logic, topos, coherent topos, ultracategory, coherent logic, regular logic.  \nMSC2020 . 03B10, 03G30, 18B25, 18C10, 18A15, 18F10, 18N10 .  \nContents  \n1. Introduction........................................................... 2  \n1.1. On logics and doctrines ........................................... 6  \n2. Conceptual completeness as ‘every algebra is a syntactic category’..... 8  \n2.1. Syntactic categories are TH-algebras .............................. 9  \n2.2. A Diaconescu-like adjunction: SynH ⊣ ClH ......................... 11  \n3. Four easy pieces ....................................................... 12  \n3.1. The reduction lemma .............................................. 13  \n3.2. Coherent logic ..................................................... 14  \n3.3. Regular logic ...................................................... 16  \n3.4. Essentially algebraic logic with falsum ............................. 18  \n3.5. Finitary disjunctive logic .......................................... 20  \n3.6. Other examples and remarks ...................................... 23  \n4. Conservatively embedded logics........................................ 24  \n5. It was Makkai all along ................................................ 27  \n5.1. Recalls on virtual ultracategories .................................. 29  \n5.2. Revisiting semantic prescriptions .................................. 30  \n5.3. Back to Makkai’s conceptual completeness ......................... 32  \n5.4. A modular account of conceptual completeness à la Makkai........ 33  \nReferences ................................................................. 34  \nThe first-named author was supported by the Swedish Research Council (SRC, Vetenskapsrådet) under Grant No. 2019-04545. The research has received funding from Knut and Alice Wallenbergs Foundation through the Foundation’s program for mathematics. The second-named author acknowledges financial support from the Agence Nationale de la Recherche (ANR), project ANR- 23-CE48-0012-01 .  \n2 IVAN DI LIBERTI, UMBERTO TARANTINO, AND LINGYUAN YE  \n1. Introduction  \nOverview. A traditional completeness theorem, for a logic, expresses that the class of models of any theory T in the logic is able to separate formulas not provably equivalent modulo T. In 1987, Makkai proved his celebrated (strong) conceptual completeness 1 theorem for coherent first-order logic [Mak87]: strengthening the usual completeness, his result shows that we can reconstruct the syntax of T from its semantics, conceived as the category Mod (T) of models and their homomorphisms, provided that the latter is endowed with some additional structure. In this sense, conceptual completeness is traditionally intended as a duality between syntax and set-based semantics. The essence of the t","cbCaifnmoLOJowTD","https://ap.wps.com/l/cbCaifnmoLOJowTD","pdf",887718,1,35,"English","en",105,"# Introduction\n## On logics and doctrines\n# Conceptual completeness as ‘every algebra is a syntactic category’\n## Syntactic categories are TH-algebras\n## A Diaconescu-like adjunction: SynH ⊣ ClH\n# Four easy pieces\n## The reduction lemma\n## Coherent logic\n## Regular logic\n## Essentially algebraic logic with falsum\n## Finitary disjunctive logic\n## Other examples and remarks\n# Conservatively embedded logics\n# It was Makkai all along\n## Recalls on virtual ultracategories\n## Revisiting semantic prescriptions\n## Back to Makkai’s conceptual completeness\n## A modular account of conceptual completeness à la Makkai\n# References","[{\"question\":\"What notion of conceptual completeness does the paper investigate?\",\"answer\":\"It investigates conceptual completeness for a fixed fragment of geometric logic, formulated in the authors’ framework rather than via the purely traditional syntax-from-semantics viewpoint.\"},{\"question\":\"How does the paper characterize conceptual completeness?\",\"answer\":\"It characterizes conceptual completeness using a duality between theories and topoi, translating the property into a proof-theoretic perspective.\"},{\"question\":\"What is the relationship between the paper’s notion and Makkai’s original reconstruction theorem?\",\"answer\":\"Under completeness with respect to set-based models, the paper shows the notion is equivalent to a traditional reconstruction result; in the coherent case, it recovers Makkai’s original reconstruction theorem via ultracategories.\"}]",1784182119,88,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"conceptual-completeness-for-subgeometric-logics","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/conceptual-completeness-for-subgeometric-logics/82658/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What notion of conceptual completeness does the paper investigate?","Question",{"text":75,"@type":76},"It investigates conceptual completeness for a fixed fragment of geometric logic, formulated in the authors’ framework rather than via the purely traditional syntax-from-semantics viewpoint.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper characterize conceptual completeness?",{"text":80,"@type":76},"It characterizes conceptual completeness using a duality between theories and topoi, translating the property into a proof-theoretic perspective.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the relationship between the paper’s notion and Makkai’s original reconstruction theorem?",{"text":84,"@type":76},"Under completeness with respect to set-based models, the paper shows the notion is equivalent to a traditional reconstruction result; 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