{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T07:20:20Z","timestamp":1759389620137},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,12,2]],"date-time":"2014-12-02T00:00:00Z","timestamp":1417478400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>Continuous lattices were characterised by Mart\u00edn Escard\u00f3 as precisely those objects that are Kan-injective with respect to a certain class of morphisms. In this paper we study Kan-injectivity in general categories enriched in posets. As an example, \u03c9-CPO's are precisely the posets that are Kan-injective with respect to the embeddings \u03c9 \u21aa \u03c9 + 1 and 0 \u21aa 1.<\/jats:p><jats:p>For every class<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0960129514000024_inline1\" \/><jats:tex-math>$\\mathcal{H}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>of morphisms, we study the subcategory of all objects that are Kan-injective with respect to<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0960129514000024_inline1\" \/><jats:tex-math>$\\mathcal{H}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and all morphisms preserving Kan extensions. For categories such as<jats:monospace>Top<\/jats:monospace><jats:sub>0<\/jats:sub>and<jats:monospace>Pos<\/jats:monospace>, we prove that whenever<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0960129514000024_inline1\" \/><jats:tex-math>$\\mathcal{H}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock\u2013Z\u00f6berlein monad. However, this does not generalise to proper classes, and we present a class of continuous mappings in<jats:monospace>Top<\/jats:monospace><jats:sub>0<\/jats:sub>for which Kan-injectivity does not yield a monadic category.<\/jats:p>","DOI":"10.1017\/s0960129514000024","type":"journal-article","created":{"date-parts":[[2014,12,2]],"date-time":"2014-12-02T14:43:56Z","timestamp":1417531436000},"page":"6-45","source":"Crossref","is-referenced-by-count":9,"title":["Kan injectivity in order-enriched categories"],"prefix":"10.1017","volume":"25","author":[{"given":"JI\u0158\u00cd","family":"AD\u00c1MEK","sequence":"first","affiliation":[]},{"given":"LURDES","family":"SOUSA","sequence":"additional","affiliation":[]},{"given":"JI\u0158\u00cd","family":"VELEBIL","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,12,2]]},"reference":[{"key":"S0960129514000024_ref19","first-page":"1","volume-title":"Basic concepts of enriched category theory","author":"Kelly","year":"1982"},{"key":"S0960129514000024_ref26","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0073967"},{"key":"S0960129514000024_ref27","doi-asserted-by":"publisher","DOI":"10.1137\/0211062"},{"key":"S0960129514000024_ref25","unstructured":"Reiterman J. 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