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In this paper, for any commutative quantale\n \n \n $$\\mathcal {V}$$<\/jats:tex-math>\n \n V<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n taking the role of\n \n \n $$[0,\\infty ]$$<\/jats:tex-math>\n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n \u221e<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , we extend these two characterisations of Lawvere-style completeness from ordinary to\n \n \n $$\\mathcal {V}$$<\/jats:tex-math>\n \n V<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -normed categories, that is, to categories enriched in the category of\n \n \n $$\\mathcal {V}$$<\/jats:tex-math>\n \n V<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -normed sets, i.e., of sets equipped with a\n \n \n $$\\mathcal {V}$$<\/jats:tex-math>\n \n V<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -valued function. We also establish improvements of recent results regarding the normed convergence of Cauchy sequences in two important\n \n \n $$\\mathcal {V}$$<\/jats:tex-math>\n \n V<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -normed categories.\n <\/jats:p>","DOI":"10.1007\/s10485-026-09865-z","type":"journal-article","created":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T07:34:25Z","timestamp":1777880065000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Notions of Cauchy Completeness for Normed Categories"],"prefix":"10.1007","volume":"34","author":[{"given":"Dirk","family":"Hofmann","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Walter","family":"Tholen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,4]]},"reference":[{"issue":"12","key":"9865_CR1","doi-asserted-by":"publisher","first-page":"306","DOI":"10.70930\/tac\/r1jknjot","volume":"36","author":"T Avery","year":"2021","unstructured":"Avery, T., Leinster, T.: Isbell conjugacy and the reflexive completion. 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