Coherence for bicategories and indexed categories. (English) Zbl 0567.18003

In this well-written article the authors provide a consistent approach to the subject described by the title, giving complete (partially modified or simplified) definitions, presenting the coherence results on a reasonable level of generality, carefully comparing their work with J. Bénabou’s early results [cf. Lect. Notes Math. 47, 1-77 (1967; Zbl 0165.330) and Cah. Topologie Géom. Différ. 10, 1-126 (1968; Zbl 0162.326)], and finally adapting M. L. Laplaza’s elegant approach [J. Algebra 84, 305-323 (1983; Zbl 0525.18005)] to the present context.
Reviewer: W.Tholen


18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
Full Text: DOI


[1] Bénabou, J., Introduction to Bicategories, (Reports of the Midwest Category Seminar. Reports of the Midwest Category Seminar, Lecture Notes in Math., 47 (1967), Springer: Springer Berlin) · Zbl 1375.18001
[2] Bénabou, J., Structures algébriques dans les catégories, Cahiers Topologie Géom. Différentielle, 10, 1-126 (1968) · Zbl 0162.32602
[3] Bénabou, J., Fibrations petites et localement petites, C.R. Acad. Sci. Paris, 281, 897-900 (1975) · Zbl 0349.18006
[4] Bergman, G. M., The diamond lemma for ring theory, Advances in Math., 29, 178-218 (1978) · Zbl 0326.16019
[5] Gray, J., Formal Category Theory: Adjointness for 2-Categories, (Lecture Notes in Math., 391 (1974), Springer: Springer Berlin) · Zbl 0285.18006
[6] Grothendieck, A., Revêtements Etales et Groupe Fondamental, Séminaire de Géométrie Algébrique du Bois Marie 1960/1961, SGA1, (Lecture Notes in Math., 224 (1971), Springer: Springer Berlin)
[7] Kelly, G. M., Basic Concepts of Enriched Category Theory, (London Math. Soc. Lecture Note Series, 64 (1982), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0709.18501
[8] Laplaza, M. L., Coherence for categories with group structure: an alternative approach, J. Algebra, 84, 305-323 (1983) · Zbl 0525.18005
[9] MacLane, S., Natural associativity and commutativity, Rice University Studies, 49, 28-46 (1963) · Zbl 0244.18008
[10] MacLane, S., Categories for the Working Mathematician, (Graduate Texts in Mathematics, 5 (1971), Springer: Springer Berlin) · Zbl 0705.18001
[11] Mitchell, B., Low dimensional group cohomology as monoidal structures, Amer. J. Math., 105, 1049-1066 (1983) · Zbl 0527.18005
[12] Newman, M. H.A., On theories with a combinatorial definition of “equivalence”, Ann. of Math., 43, 223-243 (1942) · Zbl 0060.12501
[13] Paré, R.; Schumacher, D., Abstract Families and the Adjoint Functor Theorems, Indexed Categories and Their Applications, (Lecture Notes in Math., 661 (1978), Springer: Springer Berlin) · Zbl 0372.00009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.