Variation through enrichment. (English) Zbl 0571.18004

The authors begin with a systematic account of the 2-category W-Cat of categories enriched over a bicategory W. The case where W has exactly one object is the classical one of categories enriched over a monoidal category; the more general case not only allows new applications but also provides new theoretical insights.
Call a bicategory W a base if each hom-category W(u,v) admits small colimits which are preserved by composition with any \(v\to v'\) and with any u’\(\to u\). Call it a small base if the set of objects of W is small, and a closed base if each W(u,v) admits small limits and the functor W(u,v)\(\to W(u,v')\) induced by any \(v\to v'\) has a right adjoint; one could similarly define ”biclosed base”. [These are not the authors’ terms in this paper, but see the third author, Rend. Semin. Mat. Fis. Milano 51, 217-233 (1981; Zbl 0538.18005).]
The authors consider a small base W, and show that W-Cat is cocomplete as a 2-category, and complete when the base W is closed. They define the bicategory W-Mod of W-modules - which are what have been variously called bimodules, or profunctors, or distributors; and show it to be again a base, closed if W is.
For any small category C we have the 2-category Fib C of fibrations over C. With such a C the authors now associate a small base \(W=W(C)\), and they define a 2-functor L: Fib \(C\to W\)-Cat which has a right adjoint and preserves small limits. Using an abstract notion of a module in a bicategory (and in particular in a 2-category), they extend L to a homomorphism of bicategories L: Mod(Fib C)\(\to Mod(W-Cat)\), and observe that the bicategory Mod(W-Cat) of modules in W-Cat is biequivalent to the W-Mod above.
A module is said to be Cauchy if it has a right adjoint, and is said to be convergent if it arises from an arrow in the bicategory; an object in the bicategory is said to be Cauchy complete if every Cauchy module into it is convergent. A W-category is shown to be Cauchy complete precisely when it is so in (the obvious extension of) the now-classical sense for enriched categories; while a fibration over C is Cauchy complete precisely when idempotents split in each fibre.
Restricting the homomorphism of bicategories L to the Cauchy-complete objects gives the main result, a biequivalence (Fib C)\({}_{cc}\sim (W\)- Cat)\({}_{cc}\), where these are the sub-2-categories determined by the Cauchy-complete objects. Cutting down from Fib C to the bicategory Rel C of relations, we refind the presheaf part of Walters’ characterization of the sheaves on a site as the Cauchy-complete W-categories for a suitable W [the last author, J. Pure Appl. Algebra 24, 95-102 (1982; Zbl 0497.18016)]. (See also the cited paper of R. Street.)


18D20 Enriched categories (over closed or monoidal categories)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18D30 Fibered categories
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
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