{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T02:27:47Z","timestamp":1769653667145,"version":"3.49.0"},"reference-count":9,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T00:00:00Z","timestamp":1760659200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T00:00:00Z","timestamp":1760659200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100012352","name":"Universit\u00e0 degli Studi di Milano","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100012352","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100019180","name":"HORIZON EUROPE European Research Council","doi-asserted-by":"publisher","award":["670624"],"award-info":[{"award-number":["670624"]}],"id":[{"id":"10.13039\/100019180","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":[[2025,12]]},"abstract":"Abstract<\/jats:title>\n \n Let\n \n $$\\mathbf{C}$$<\/jats:tex-math>\n <\/jats:inline-formula>\n be a small category. The subtoposes of\n \n $$[\\mathbf{C}^\\textrm{op},\\mathbf{Set}]$$<\/jats:tex-math>\n <\/jats:inline-formula>\n are sometimes all of the form\n \n $$[\\mathbf{D}^\\textrm{op},\\mathbf{Set}]$$<\/jats:tex-math>\n <\/jats:inline-formula>\n where\n \n $$\\mathbf{D}$$<\/jats:tex-math>\n <\/jats:inline-formula>\n is a full subcategory of\n \n $$\\mathbf{C}$$<\/jats:tex-math>\n <\/jats:inline-formula>\n . This is the case for instance when\n \n $$\\mathbf{C}$$<\/jats:tex-math>\n <\/jats:inline-formula>\n is Cauchy-complete and finite, an Artinian poset, or the simplex category. We call such a category\n universally rigid<\/jats:italic>\n . A universally rigid category whose slices are also universally rigid, such as the aforementioned examples, is called\n stably universally rigid<\/jats:italic>\n . We provide two equivalent characterizations of such categories. The first one stipulates the existence of a winning strategy in a two-player game, and the second one combines two \u201clocal\u201d properties of\n \n $$\\mathbf{C}$$<\/jats:tex-math>\n <\/jats:inline-formula>\n involving respectively the poset reflections of its slices and its endomorphism monoids.\n <\/jats:p>","DOI":"10.1007\/s10485-025-09833-z","type":"journal-article","created":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T09:33:13Z","timestamp":1760693593000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Criterion for Categories on Which Every Grothendieck Topology is Rigid"],"prefix":"10.1007","volume":"33","author":[{"given":"J\u00e9r\u00e9mie","family":"Marqu\u00e8s","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,17]]},"reference":[{"issue":"2","key":"9833_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2023.103388","volume":"175","author":"S van Goolvan Gool","year":"2024","unstructured":"van Goolvan Gool, S., Marqu\u00e8s, J.: On duality and model theory for polyadic spaces. Ann. Pure Appl. Logic 175(2), 103388 (2024)","journal-title":"Ann. Pure Appl. Logic"},{"key":"9833_CR2","series-title":"Oxford Logic Guides","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 Logic Guides, Oxford University Press, Oxford (2002)"},{"key":"9833_CR3","series-title":"Memoirs of the American Mathematical Society","volume-title":"An Extension of the Galois Theory of Grothendieck","author":"A Joyal","year":"1984","unstructured":"Joyal, A., Tierney, M.: An Extension of the Galois Theory of Grothendieck. Memoirs of the American Mathematical Society, vol. 51. American Mathematical Society, Providence, RI (1984)"},{"issue":"2","key":"9833_CR4","first-page":"289","volume":"41","author":"GM Kelly","year":"1989","unstructured":"Kelly, G.M., Lawvere, F.W.: On the complete lattice of essential localizations. Bull. Soc. Math. Belg. S\u00e9r. A 41(2), 289\u2013319 (1989)","journal-title":"Bull. Soc. Math. Belg. S\u00e9r. A"},{"key":"9833_CR5","unstructured":"Lindenhovius, B.: Grothendieck topologies on a poset (2014). arxiv:1405.4408"},{"key":"9833_CR6","unstructured":"Marqu\u00e8s, J.: Categorical logic from the perspective of duality and compact ordered spaces. PhD thesis, Universit\u00e9 C\u00f4te d\u2019Azur (2023)"},{"issue":"25","key":"9833_CR7","first-page":"714","volume":"34","author":"M Menni","year":"2019","unstructured":"Menni, M.: Monic skeleta, boundaries, aufhebung, and the meaning of \u2018one-dimensionality\u2019. Theory Appl. Categor. 34(25), 714\u2013735 (2019)","journal-title":"Theory Appl. Categor."},{"issue":"03","key":"9833_CR8","first-page":"1337","volume":"152","author":"M Menni","year":"2024","unstructured":"Menni, M.: The successive dimension, without elegance. Proc. Am. Math. Soc. 152(03), 1337\u20131354 (2024)","journal-title":"Proc. Am. Math. Soc."},{"key":"9833_CR9","unstructured":"Wrigley, J.L.: Doctrinal and groupoidal representations of classifying topoi. PhD thesis, Universit\u00e0 degli Studi dell\u2019Insubria (2024)"}],"container-title":["Applied Categorical Structures"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-025-09833-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10485-025-09833-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-025-09833-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T14:27:00Z","timestamp":1769610420000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10485-025-09833-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,17]]},"references-count":9,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["9833"],"URL":"https:\/\/doi.org\/10.1007\/s10485-025-09833-z","relation":{},"ISSN":["0927-2852","1572-9095"],"issn-type":[{"value":"0927-2852","type":"print"},{"value":"1572-9095","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,17]]},"assertion":[{"value":"6 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 October 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"37"}}