{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T01:56:38Z","timestamp":1778291798846,"version":"3.51.4"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2009,3,4]],"date-time":"2009-03-04T00:00:00Z","timestamp":1236124800000},"content-version":"unspecified","delay-in-days":5847,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[1993,3]]},"abstract":"<jats:p>The notion of<jats:italic>semi-functor<\/jats:italic>was introduced in Hayashi (1985) in order to make possible a category-theoretical characterization of models of the non-extensional typed lambda calculus. Motivated by the further use of semi-functors in Martini (1987), Jacobs (1991) and Hoofman (1992a), (1992b) and (1992c), we consider the general theory of semi-functors in this paper. It turns out that the notion of<jats:italic>semi natural transformation<\/jats:italic>plays an important part in this theory, and that various categorical notions involving semi-functors can be viewed as 2-categorical notions in the 2-category of categories, semi-functors and semi natural transformations. In particular, we find that the notion of<jats:italic>normal semi-adjunction<\/jats:italic>as defined in Hayashi (1985) is the canonical generalization of the notion of adjunction to the world of semi-functors. Further topics covered in this paper are the relation between semi-functors and splittings, the Karoubi envelope construction, semi-comonads, and a semi-adjoint functor theorem.<\/jats:p>","DOI":"10.1017\/s096012950000013x","type":"journal-article","created":{"date-parts":[[2009,3,4]],"date-time":"2009-03-04T09:01:47Z","timestamp":1236157307000},"page":"93-128","source":"Crossref","is-referenced-by-count":10,"title":["The theory of semi-functors"],"prefix":"10.1017","volume":"3","author":[{"given":"Raymond","family":"Hoofman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2009,3,4]]},"reference":[{"key":"S096012950000013X_ref019","doi-asserted-by":"publisher","DOI":"10.4064\/cm-48-2-153-165"},{"key":"S096012950000013X_ref018","first-page":"399","article-title":"Categorical, Functional and Algebraic Aspects of the Type-Free Lambda Calculus","volume":"9","author":"Obtu\u0142owicz","year":"1982","journal-title":"Universal Algebra and Applications"},{"key":"S096012950000013X_ref016","first-page":"219","volume":"283","author":"Martini","year":"1987","journal-title":"An Interval Model for Second Order Lambda Calculus"},{"key":"S096012950000013X_ref008","unstructured":"Hoofman R. 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