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Doctrines whose structure forms a fully faithful adjoint string. (English) Zbl 0878.18004

The “KZ-doctrines” studied here are, in the simplest cases, monads where structure maps are adjoint to units, in the sense of the reviewer’s [“Monads for which structures are adjoint to units”, J. Pure Appl. Algebra 104, No. 1, 41-59 (1995; Zbl 0849.18008)]. They live, however, in a more lax form, in the context of Gray-categories [as considered by R. Gordon, A. J. Power and R. Street, “Coherence for tricategories, Mem. Am. Math. Soc. 558 (1995; Zbl 0836.18001)]. The author lifts the basic theory into this, very general, setting (which is also more general than that of V. Zöberlein [“Doctrines on 2-categories”, Math. Z. 148, 267-279 (1976; Zbl 0311.18005)]; no comparison with the latter is attempted).
Reviewer: A.Kock (Aarhus)

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
18D20 Enriched categories (over closed or monoidal categories)
18C20 Eilenberg-Moore and Kleisli constructions for monads
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