{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T11:03:27Z","timestamp":1784113407590,"version":"3.55.0"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T00:00:00Z","timestamp":1779148800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T00:00:00Z","timestamp":1779148800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"publisher","award":["101007627"],"award-info":[{"award-number":["101007627"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1007\/s10485-026-09872-0","type":"journal-article","created":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T15:18:10Z","timestamp":1779203890000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["From Coextensive Varieties to the Gaeta Topos"],"prefix":"10.1007","volume":"34","author":[{"given":"William","family":"Zuluaga","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,19]]},"reference":[{"key":"9872_CR1","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s00012-013-0220-9","volume":"69","author":"M Badano","year":"2013","unstructured":"Badano, M., Vaggione, D.: Varieties with equationally definable factor congruences. Algebra Universalis 69, 139\u2013166 (2013)","journal-title":"Algebra Universalis"},{"key":"9872_CR2","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1142\/S0218196716500211","volume":"26","author":"M Badano","year":"2016","unstructured":"Badano, M., Vaggione, D.: Equational definability of (complementary) central elements. Internat. J. Algebra Comput. 26, 509\u2013532 (2016)","journal-title":"Internat. J. Algebra Comput."},{"key":"9872_CR3","unstructured":"Broodryk, D.N.: Characterization of coextensive varieties of universal algebras II. arXiv preprint (2021). arXiv:2104.12188"},{"issue":"2","key":"9872_CR4","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/S0747-7171(08)80089-2","volume":"13","author":"S Burris","year":"1992","unstructured":"Burris, S.: Discriminator varieties and symbolic computation. J. Symb. Comput. 13(2), 175\u2013207 (1992)","journal-title":"J. Symb. Comput."},{"key":"9872_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8130-3","volume-title":"A Course in Universal Algebra","author":"S Burris","year":"1981","unstructured":"Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981)"},{"issue":"2","key":"9872_CR6","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/0022-4049(93)90035-R","volume":"84","author":"A Carboni","year":"1993","unstructured":"Carboni, A., Lack, S., Walters, R.F.C.: Introduction to extensive and distributive categories. J. Pure Appl. Algebra 84(2), 145\u2013158 (1993)","journal-title":"J. Pure Appl. Algebra"},{"issue":"1\u20132","key":"9872_CR7","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/S0022-4049(01)00016-0","volume":"161","author":"A Carboni","year":"2001","unstructured":"Carboni, A., Pedicchio, M.C., Rosick\u00fd, J.: Syntactic characterizations of various classes of locally presentable categories. J. Pure Appl. Algebra 161(1\u20132), 65\u201390 (2001)","journal-title":"J. Pure Appl. Algebra"},{"key":"9872_CR8","doi-asserted-by":"publisher","first-page":"3533","DOI":"10.1016\/j.jpaa.2016.04.014","volume":"220","author":"JL Castiglioni","year":"2016","unstructured":"Castiglioni, J.L., Menni, M., Zuluaga Botero, W.J.: A representation theorem for integral rigs and its applications to residuated lattices. J. Pure Appl. Algebra 220, 3533\u20133566 (2016)","journal-title":"J. Pure Appl. Algebra"},{"key":"9872_CR9","doi-asserted-by":"crossref","unstructured":"Diers, Y.: Categories of Boolean sheaves of simple algebras. In: Categories of Boolean Sheaves of Simple Algebras, pp. 48\u2013113. Springer, Berlin (2006)","DOI":"10.1007\/BFb0075890"},{"issue":"10","key":"9872_CR10","first-page":"208","volume":"4","author":"R Gates","year":"1998","unstructured":"Gates, R.: On generic separable objects. Theory Appl. Categ. 4(10), 208\u2013248 (1998)","journal-title":"Theory Appl. Categ."},{"issue":"10","key":"9872_CR11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2020.106381","volume":"224","author":"M Hoefnagel","year":"2020","unstructured":"Hoefnagel, M.: M-coextensive objects and the strict refinement property. J. Pure Appl. Algebra 224(10), 106381 (2020). https:\/\/doi.org\/10.1016\/j.jpaa.2020.106381","journal-title":"J. Pure Appl. Algebra"},{"issue":"21","key":"9872_CR12","first-page":"617","volume":"44","author":"M Hoefnagel","year":"2025","unstructured":"Hoefnagel, M., Theart, E.: On extensivity and coextensivity of morphisms. Theory Appl. Categ. 44(21), 617\u2013642 (2025)","journal-title":"Theory Appl. Categ."},{"key":"9872_CR13","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198534259.001.0001","volume-title":"Sketches of an Elephant: A Topos Theory Compendium","author":"PT Johnstone","year":"2002","unstructured":"Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium. Oxford University Press, Oxford (2002)"},{"key":"9872_CR14","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1007\/BF02488038","volume":"9","author":"J Koll\u00e1r","year":"1979","unstructured":"Koll\u00e1r, J.: Mailbox congruences and one element subalgebras. Algebra Universalis 9, 266\u2013267 (1979)","journal-title":"Algebra Universalis"},{"key":"9872_CR15","doi-asserted-by":"crossref","unstructured":"Lawvere, F.W.: Functorial semantics of algebraic theories. PhD thesis, Columbia University (1963)","DOI":"10.1073\/pnas.50.5.869"},{"issue":"14","key":"9872_CR16","first-page":"497","volume":"20","author":"FW Lawvere","year":"2008","unstructured":"Lawvere, F.W.: Core varieties, extensivity, and rig geometry. Theory Appl. Categ. 20(14), 497\u2013503 (2008)","journal-title":"Theory Appl. Categ."},{"key":"9872_CR17","volume-title":"Sheaves in Geometry and Logic: A First Introduction to Topos Theory","author":"S Mac Lane","year":"2012","unstructured":"Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, New York (2012)"},{"key":"9872_CR18","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1016\/j.jalgebra.2017.09.035","volume":"494","author":"P Mayr","year":"2018","unstructured":"Mayr, P., Ru\u0161kuc, N.: Finiteness properties of direct products of algebraic structures. J. Algebra 494, 167\u2013187 (2018)","journal-title":"J. Algebra"},{"key":"9872_CR19","volume-title":"Algebras, Lattices, Varieties","author":"R McKenzie","year":"1987","unstructured":"McKenzie, R., McNulty, G., Taylor, W.: Algebras, Lattices, Varieties, vol. 1. Wadsworth & Brooks\/Cole Math. Series, Monterey, CA (1987)"},{"issue":"4","key":"9872_CR20","first-page":"451","volume":"62","author":"M Menni","year":"2021","unstructured":"Menni, M.: A basis theorem for 2-rigs and rig geometry. Cahiers Topologie G\u00e9om. Diff\u00e9rentielle Cat\u00e9g. 62(4), 451\u2013490 (2021)","journal-title":"Cahiers Topologie G\u00e9om. Diff\u00e9rentielle Cat\u00e9g."},{"key":"9872_CR21","doi-asserted-by":"publisher","first-page":"5061","DOI":"10.1090\/S0002-9947-09-04921-6","volume":"361","author":"P Sanchez Terraf","year":"2009","unstructured":"Sanchez Terraf, P., Vaggione, D.: Varieties with definable factor congruences. Trans. Am. Math. Soc. 361, 5061\u20135088 (2009)","journal-title":"Trans. Am. Math. Soc."},{"key":"9872_CR22","unstructured":"Terraf, P.S.: Congruencias factor definibles. PhD thesis, Universidad Nacional de C\u00f3rdoba (2007)"},{"key":"9872_CR23","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1007\/BF01233918","volume":"36","author":"D Vaggione","year":"1996","unstructured":"Vaggione, D.: Varieties of shells. Algebra Universalis 36, 483\u2013487 (1996)","journal-title":"Algebra Universalis"},{"key":"9872_CR24","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1006\/jabr.1996.0268","volume":"184","author":"D Vaggione","year":"1996","unstructured":"Vaggione, D.: Varieties in which the pierce stalks are directly indecomposable. J. Algebra 184, 424\u2013434 (1996)","journal-title":"J. Algebra"},{"issue":"2","key":"9872_CR25","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1006\/aima.1999.1855","volume":"148","author":"D Vaggione","year":"1999","unstructured":"Vaggione, D.: Central elements in varieties with the fraser - horn property. Adv. Math. 148(2), 193\u2013202 (1999)","journal-title":"Adv. Math."},{"key":"9872_CR26","unstructured":"Zanfa, R.: Classifying topoi for extensions of algebraic theories with indecomposable models. Msc thesis, Universit\u00e0 degli studi di Milano (2017)"},{"key":"9872_CR27","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1007\/s00012-021-00745-2","volume":"82","author":"WJ Zuluaga Botero","year":"2021","unstructured":"Zuluaga Botero, W.J.: Coextensive varieties via central elements. Algebra Universalis 82, 50 (2021)","journal-title":"Algebra Universalis"}],"container-title":["Applied Categorical Structures"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-026-09872-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10485-026-09872-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-026-09872-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T10:03:36Z","timestamp":1784109816000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10485-026-09872-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,19]]},"references-count":27,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["9872"],"URL":"https:\/\/doi.org\/10.1007\/s10485-026-09872-0","relation":{},"ISSN":["0927-2852","1572-9095"],"issn-type":[{"value":"0927-2852","type":"print"},{"value":"1572-9095","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,19]]},"assertion":[{"value":"31 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 May 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 May 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"This project has been funded by the MOSAIC Project 101007627, within the European Union\u2019s Horizon 2020 research and innovation programme under the Marie Sk\u0142odowska-Curie Actions.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","label":"Ethical Approval","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"36"}}