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Enriched categories, internal categories and change of base. (English) Zbl 1254.18001

This paper is a reprint of the author’s 1992 thesis at the University of Cambridge. It has not been published except as Macquarie Mathematics Reports 93–123/1 and 93–123/2 (Macquarie University, 1993). The influence of the thesis over the years certainly warrants making it widely available to the growing number of researchers using these ideas.
In the reprinting, the author has rephrased the Introduction somewhat, without changing the structure and content. He has added an Epilogue which explains how some more recent developments fit with this work.
Unified by the goal of providing a machine to produce the structure common to enriched categories and to internal categories, the thesis contains many quite distinct and original achievements. Let us look at these.
(a)
There is a precise unit-counit definition of local adjunction between a morphism \(G:\mathcal{B} \to \mathcal{C}\) and a comorphism \(F:\mathcal{C} \to \mathcal{B}\) of bicategories \(\mathcal{B}\) and \(\mathcal{C}\).
(b)
The multicategory structure on the category \(\mathbf{Hom}\) of bicategories and homomorphisms is analysed.
(c)
Categories enriched in \(\mathbf{Hom}\) are called bicategory enriched (as distinct from categories enriched in a bicategory in the sense of R. F. C. Walters [“Sheaves on sites as Cauchy-complete categories”, J. Pure Appl. Algebra 24, 95–102 (1982; Zbl 0497.18016)]). The theory of these is developed including the notion of biadjunction within one.
(d)
Double bicategories are defined and shown to form the objects of a bicategory enriched category \(\mathbf{Horiz}\).
(e)
A theorem is proved on the existence of a biadjoint for a morphism \(G:\mathcal{B} \to \mathcal{C}\) in \(\mathbf{Horiz}\). A refinement is given for the case where \(\mathcal{B}\) and \(\mathcal{C}\) are equipments (in a sense arising from R. J. Wood [“Abstract proarrows. I”, Cah. Topol. Géom. Différ. 23, 279–290 (1982; Zbl 0497.18012)]).
(f)
Item (e) is applied to yield change of base theorems for matrices over a bicategory, spans in a finitely complete category, and relations in a regular category.
(g)
The equipment of monads construction is exhibited as a bicategory enriched functor which applies to the examples in item (f) to yield change of base theorems for categories enriched in a bicategory, for categories in a finitely complete category, and for order ideals in a regular category.
(h)
A theorem is proved on the behaviour of weighted colimits under change of base.
(i)
There is a “comparison lemma” for sites on bicategories which generalises those for usual sheaves and stacks.
(j)
The machinery is applied to complete Paré’s project to “explain” weighted colimits in 2-categories in terms of double categories.
(k)
The original pasting in bicategories used by J. Bénabou [“Introduction to bicategories”, Lect. Notes Math. 47, 1–77 (1967; Zbl 0165.33001)] is put on a firm foundation.
This work exhibits a mastery of a wide range of topics from category theory and extends definitively various disparate works. The author does not balk at introducing new structures when they are the appropriate conceptual level for the matter at hand. The genius of the thesis is in finding the manageable cloak to cover the body of topics.

MSC:

18-02 Research exposition (monographs, survey articles) pertaining to category theory
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18D20 Enriched categories (over closed or monoidal categories)
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