{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T09:24:11Z","timestamp":1648632251315},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2008,10,1]],"date-time":"2008-10-01T00:00:00Z","timestamp":1222819200000},"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":[[2008,10]]},"abstract":"<jats:p>Given a map<jats:italic>f<\/jats:italic>in the category \u03c9-<jats:bold>Cpo<\/jats:bold>of \u03c9-complete posets, exponentiability of<jats:italic>f<\/jats:italic>in \u03c9-<jats:bold>Cpo<\/jats:bold>easily implies exponentiability of<jats:italic>f<\/jats:italic>in the category<jats:bold>Pos<\/jats:bold>of posets, while the converse is not true. We investigate the extra conditions needed on<jats:italic>f<\/jats:italic>exponentiable in<jats:bold>Pos<\/jats:bold>to be exponentiable in \u03c9-<jats:bold>Cpo<\/jats:bold>by showing the existence of partial products of the two-point ordered set<jats:italic>S<\/jats:italic>={0&lt;1} (Theorem 2.8). Using this characterisation and the embedding through the Scott topology of \u03c9-<jats:bold>Cpo<\/jats:bold>in the category<jats:bold>Top<\/jats:bold>of topological spaces, we compare exponentiability in each setting and find that a morphism in \u03c9-<jats:bold>Cpo<\/jats:bold>that is exponentiable in both<jats:bold>Top<\/jats:bold>and<jats:bold>Pos<\/jats:bold>is exponentiable in \u03c9-<jats:bold>Cpo<\/jats:bold>also. Furthermore, we show that the exponentiability in<jats:bold>Top<\/jats:bold>and<jats:bold>Pos<\/jats:bold>are independent of each other.<\/jats:p>","DOI":"10.1017\/s0960129508006786","type":"journal-article","created":{"date-parts":[[2008,5,28]],"date-time":"2008-05-28T13:32:13Z","timestamp":1211981533000},"page":"1005-1016","source":"Crossref","is-referenced-by-count":1,"title":["Exponentiable morphisms of domains"],"prefix":"10.1017","volume":"18","author":[{"given":"F.","family":"CAGLIARI","sequence":"first","affiliation":[]},{"given":"S.","family":"MANTOVANI","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2008,10,1]]},"reference":[{"key":"S0960129508006786_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(01)00046-9"},{"key":"S0960129508006786_ref12","first-page":"16","article-title":"Exponentiable morphisms: posets, spaces, locales, and Grothendieck toposes","volume":"8","author":"Niefield","year":"2001","journal-title":"Theory and Applications of Categories"},{"key":"S0960129508006786_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02485815"},{"key":"S0960129508006786_ref8","article-title":"Methode de la descente","volume":"2","author":"Giraud","year":"1964","journal-title":"Bull. Soc. Math. France, Memoire"},{"key":"S0960129508006786_ref6","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1017\/S0960129500070031","article-title":"Colimits in the category DCPO","volume":"6","author":"Fiech","year":"1996","journal-title":"Mathematical Structures in Computer Science"},{"key":"S0960129508006786_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(87)90124-1"},{"key":"S0960129508006786_ref3","first-page":"139","article-title":"The convergence approach to exponentiable maps","volume":"60","author":"Clementino","year":"2003","journal-title":"Port. Math. (N.S.)"},{"key":"S0960129508006786_ref15","first-page":"301","article-title":"Subspaces In Abstract Stone Duality","volume":"10","author":"Taylor","year":"2002","journal-title":"Theory and Applications of Categories"},{"key":"S0960129508006786_ref10","doi-asserted-by":"publisher","DOI":"10.1147\/rd.202.0138"},{"key":"S0960129508006786_ref1","first-page":"1","article-title":"Domain theory","volume":"3","author":"Abramsky","year":"1994","journal-title":"Handbook of logic in computer science"},{"key":"S0960129508006786_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0073967"},{"key":"S0960129508006786_ref7","volume-title":"Encyclopedia of Mathematics and its Applications","author":"Gierz","year":"2003"},{"key":"S0960129508006786_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(82)90004-4"},{"key":"S0960129508006786_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2007.02.004"},{"key":"S0960129508006786_ref2","volume-title":"Abstract and Concrete Categories","author":"Ad\u00e1mek","year":"1990"},{"key":"S0960129508006786_ref16","unstructured":"Tholen W. (2000) Injectives, exponentials, and model categories. In: Abstracts of the Int. Conf. on Category Theory (Como, Italy, 2000) 183\u2013190."}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129508006786","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,6]],"date-time":"2020-05-06T00:12:24Z","timestamp":1588723944000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129508006786\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,10]]},"references-count":16,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2008,10]]}},"alternative-id":["S0960129508006786"],"URL":"https:\/\/doi.org\/10.1017\/s0960129508006786","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,10]]}}}