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**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.)

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.)

### MSC:

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) |

### Keywords:

2-category; bicategory; monoidal category; small base; closed base; fibrations; Cauchy module; enriched categories; Cauchy complete; sheaves on a site; W-categories
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\textit{R. Betti} et al., J. Pure Appl. Algebra 29, 109--127 (1983; Zbl 0571.18004)

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### References:

[1] | Bénabou, J., Introduction to bicategories, (Lecture Notes in Math., 47 (1967), Springer: Springer Berlin), 1-77 · Zbl 1375.18001 |

[2] | Bénabou, J., Les distributeurs, (Séminaires de Math. Pure (1973), Univ. Catholique de Louvain), Rapport No. 33 · Zbl 0162.32602 |

[3] | Betti, R.; Carboni, A., Cauchy completion and the associated sheaf, Cahiers de Topologie et Géom. Diff., 23, 243-256 (1982) · Zbl 0496.18008 |

[4] | Betti, R.; Carboni, A., Notion of topology for bicategories, Cahiers de Topologie et Géom. Diff. (1983), to appear. · Zbl 0513.18007 |

[5] | Kelly, G. M.; Street, R. H., Review of the elements of 2-categories, (Lecture Notes in Math., 420 (1974), Springer: Springer Berlin), 75-103 |

[6] | Lawvere, F. W., Closed categories and biclosed bicategories, Lectures at Math. Inst. Aarhus Universitet (Fall, 1971) |

[7] | Linton, F. E.J., Coequalizers in categories of algebras, (Lecture Notes in Math., 80 (1969), Springer: Springer Berlin), 75-90 · Zbl 0181.02902 |

[8] | MacLane, S., Categories for the Working Mathematician (1971), Springer: Springer New York-Heidelberg-Berlin |

[9] | Street, R. H., The formal theory of monads, J. Pure Appl. Algebra, 2, 149-168 (1972) · Zbl 0241.18003 |

[10] | Street, R. H., Fibrations in bicategories, Cahiers de Topologie et Géom. Diff., 21, 111-160 (1980) · Zbl 0436.18005 |

[11] | Street, R. H., Conspectus of variable categories, J. Pure Appl. Algebra, 21, 307-338 (1981) · Zbl 0469.18007 |

[12] | Street, R. H., Enriched categories and cohomology, Quaestiones Math. (1983), to appear. · Zbl 0523.18007 |

[13] | Street, R. H.; Walters, R. F.C., Yoneda structures on 2-categories, J. Algebra, 50, 350-379 (1978) · Zbl 0401.18004 |

[14] | Walters, R. F.C., Sheaves on sites as Cauchy-complete categories, J. Pure Appl. Algebra, 24, 95-102 (1982) · Zbl 0497.18016 |

[15] | Wolff, H., x-cat and x-graph, J. Pure Appl. Algebra, 4, 123-135 (1974) · Zbl 0282.18010 |

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