{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T04:24:07Z","timestamp":1777350247858,"version":"3.51.4"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3479,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,9]]},"abstract":"Let us say that a geometric theory T<\/jats:italic> is of presheaf type<\/jats:italic> if its classifying topos is (equivalent to) a presheaf topos. (We adhere to the convention that geometric logic<\/jats:italic> allows arbitrary disjunctions, while coherent logic<\/jats:italic> means geometric and finitary.) Write Mod(T<\/jats:italic>) for the category of Set<\/jats:italic>-models and homomorphisms of T<\/jats:italic>. The next proposition is well known; see, for example, MacLane\u2013Moerdijk [13], pp. 381-386, and the textbook of Ad\u00e1mek\u2013Rosick\u00fd [1] for additional information:<\/jats:p>Proposition 0.1. For a category , the following properties are equivalent<\/jats:italic>:<\/jats:p>(i) is a finitely accessible category in the sense of Makkai\u2013Par\u00e9<\/jats:italic> [14], i.e., it has filtered colimits and a small dense subcategory of finitely presentable objects<\/jats:italic><\/jats:p>ii) is equivalent to Pts, the category of points of some presheaf topos<\/jats:italic><\/jats:p>(iii) is equivalent to the free filtered cocompletion (also known as Ind-) of a small category <\/jats:italic>.<\/jats:p>(iv) is equivalent to<\/jats:italic> Mod(T<\/jats:italic>) for some geometric theory of presheaf type<\/jats:italic>.<\/jats:p>Moreover, if these are satisfied for a given , then the \u2014in any of<\/jats:italic> (i), (ii) and<\/jats:italic> (iii)\u2014can be taken to be the full subcategory of consisting of finitely presentable objects. (There may be inequivalent choices of , as it is in general only determined up to idempotent completion; this will not concern us.)<\/jats:italic><\/jats:p>This seems to completely solve the problem of identifying when T<\/jats:italic> is of presheaf type: check whether Mod(T<\/jats:italic>) is finitely accessible and if so, recover the presheaf topos as Set<\/jats:italic>-functors on the full subcategory of finitely presentable models. There is a subtlety here, however, as pointed out (probably for the first time) by Johnstone [10].<\/jats:p>","DOI":"10.2178\/jsl\/1096901776","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:46:25Z","timestamp":1109799985000},"page":"923-934","source":"Crossref","is-referenced-by-count":12,"title":["Theories of presheaf type"],"prefix":"10.1017","volume":"69","author":[{"given":"Tibor","family":"Beke","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007726_ref018","first-page":"281","volume-title":"Category theoretic methods in geometry","author":"Wraith","year":"1983"},{"key":"S0022481200007726_ref017","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-339050-9.50021-3"},{"key":"S0022481200007726_ref016","volume-title":"Cyclic sets as a classifying topos","author":"Moerdijk","year":"1995"},{"key":"S0022481200007726_ref014","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/104"},{"key":"S0022481200007726_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-85844-4"},{"key":"S0022481200007726_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061820"},{"key":"S0022481200007726_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-38117-4"},{"key":"S0022481200007726_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(80)90040-7"},{"key":"S0022481200007726_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600579"},{"key":"S0022481200007726_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0927-0"},{"key":"S0022481200007726_ref007","first-page":"207","volume-title":"Orders: description and roles","volume":"99","author":"Hodges","year":"1984"},{"key":"S0022481200007726_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8707-6"},{"key":"S0022481200007726_ref009","volume-title":"Topos theory","volume":"10","author":"Johnstone","year":"1977"},{"key":"S0022481200007726_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/030\/749771"},{"key":"S0022481200007726_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061828"},{"key":"S0022481200007726_ref011","volume-title":"Stone spaces","author":"Johnstone","year":"1982"},{"key":"S0022481200007726_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066201"},{"key":"S0022481200007726_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(69)90050-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007726","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T20:27:35Z","timestamp":1557174455000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007726\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,9]]},"references-count":18,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2004,9]]}},"alternative-id":["S0022481200007726"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1096901776","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,9]]}}}