# zbMATH — the first resource for mathematics

An axiomatics for bicategories of modules. (English) Zbl 0615.18006
The reviewer [Rend. Semin. Mat. Fis. Milano 51, 217-233 (1983; Zbl 0538.18005)] called a bicategory W a base when it had a small set of objects, each hom-category had small colimits and composition on either side with an arrow preserved small colimits. He gave a characterization of the bicategory W-Mod of small categories with homs enriched in W and modules between them. Bases should be compared with the sites of topos theory in the light of R. F. C. Walters [J. Pure Appl. Algebra 24, 95-102 (1982; Zbl 0497.18016)].
The present paper considers W-Mod freed from the restriction that W have a small set of objects. Then one can consider (W-Mod)-Mod which is shown to be biequivalent to W-Mod. This result compares with the fact that the category of sheaves for the canonical topology on a topos is equivalent to the original topos. Their result bears the same relationship to the reviewer’s characterization as the result of the last sentence does to Giraud’s topos-characterization theorem.
Reviewer: R.H.Street

##### MSC:
 18D20 Enriched categories (over closed or monoidal categories) 16B50 Category-theoretic methods and results in associative algebras (except as in 16D90) 18D30 Fibered categories 18D05 Double categories, $$2$$-categories, bicategories and generalizations (MSC2010)
Full Text:
##### References:
 [1] Bénabou, J., Introduction to bicategories, (), 1-77 · Zbl 1375.18001 [2] Betti, R., Bicategorie di base, Istituto matematico milano, 2/S, II, (1980) [3] Betti, R.; Carboni, A., Cauchy completion and the associated sheaf, Cahiers topologie Géom. différentielle, 23, 243-256, (1982) · Zbl 0496.18008 [4] Betti, R.; Carboni, A.; Street, R.; Walters, R.F.C., Variation through enrichment, J. pure appl. algebra, 29, 109-127, (1983) · Zbl 0571.18004 [5] R. Betti and R.F.C. Walters, Closed bicategories and variable category theory, to appear. · Zbl 0498.18007 [6] Carboni, A.; Kasangian, S.; Walters, R.F.C., Some basic facts about bicategories and modules, Dipartimento di matematica, milano, 6, (1985) [7] A. Carboni and R. Street, Order ideals in categories, to appear. · Zbl 0565.18001 [8] Johnstone, P., Topos theory, (1977), Academic Press New York · Zbl 0368.18001 [9] S. Kasangian and R.F.C. Walters, An abstract notion of glueing, Preprint. [10] Kelly, G.M., Basic concepts of enriched category theory, (1982), Cambridge University Press Cambridge · Zbl 0478.18005 [11] Kelly, G.M.; Street, R., Review of the elements 2-categories, (), 75-103 [12] Lawvere, F.W., Closed categories and biclosed bicategories, Lectures at Aarhus university, (1971) · Zbl 0261.18010 [13] Rosebrugh, R.D.; Wood, R.J., Cofibrations in the bicategory of topoi, J. pure appl. algebra, 32, 71-94, (1984) · Zbl 0535.18006 [14] Street, R., Fibrations in bicategories, Cahiers topologie Géom. différentielle, 21, 111-160, (1980) · Zbl 0436.18005 [15] Street, R., Enriched categories and cohomology, Quaestiones math., 6, 265-283, (1983) · Zbl 0523.18007 [16] Street, R., Cauchy characterization of enriched categories, Rend. sem. mat. fis. milano, LI, 217-233, (1983) · Zbl 0538.18005 [17] Street, R.; Walters, R.F.C., Yoneda structures on 2-categories, J. algebra, 350-379, (1978) · Zbl 0401.18004 [18] Walters, R.F.C., Sheaves and Cauchy complete categories, Cahiers topologie Géom. différentielle, 22, 282-286, (1981) · Zbl 0495.18009 [19] Walters, R.F.C., Sheaves on sites as Cauchy complete categories, J. pure appl. algebra, 24, 95-102, (1982) · Zbl 0497.18016 [20] Wood, R.J., Abstract proarrows I, Cahiers topologie Géom. différentielle, 23, 279-290, (1982) · Zbl 0497.18012 [21] R.J. Wood, Proarrows II, to appear.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.