## Notes on enriched categories with colimits of some class

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Venue: | Theory Appl. Categ |

Citations: | 9 - 0 self |

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@ARTICLE{Kelly_noteson,

author = {G. M. Kelly and V. Schmitt},

title = {Notes on enriched categories with colimits of some class},

journal = {Theory Appl. Categ},

year = {}

}

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### Abstract

The paper is in essence a survey of categories having φ-weighted colimits for all the weights φ in some class Φ. We introduce the class Φ + of Φ-flat weights which are those ψ for which ψ-colimits commute in the base V with limits having weights in Φ; and the class Φ − of Φ-atomic weights, which are those ψ for which ψ-limits commute in the base V with colimits having weights in Φ. We show that both these classes are saturated (that is, what was called closed in the terminology of [AK88]). We prove that for the class P of all weights, the classes P + and P − both coincide with the class Q of absolute weights. For any class Φ and any category A, we have the free Φ-cocompletion Φ(A) of A; and we recognize Q(A) as the Cauchy-completion of A. We study the equivalence between (Q(A op)) op and Q(A), which we exhibit as the restriction of the Isbell adjunction between [A, V] op and [A op, V] when A is small; and we give a new Morita theorem for any class Φ containing Q. We end with the study of Φ-continuous weights and their relation to the Φ-flat weights. 1

### Citations

377 | Basic concepts of enriched category theory, volume 64
- Kelly
- 1982
(Show Context)
Citation Context ...he improvement of our exposition; we thankfully acknowledge their help. 2 Revision of weighted limits and colimits The necessary background knowledge about enriched categories is largely contained in =-=[Kel82]-=-, augmented by [Kel82-2] and the Albert-Kelly article [AK88]. We deal with categories enriched in a symmetric monoidal closed category V, supposing as usual that the ordinary category V0 underlying V ... |

196 |
Locally Presentable and Accessible Categories
- Adámek, Rosicky
- 1994
(Show Context)
Citation Context ...6]. These authors were concerned with extending to the enriched case the notion of accessible category and its properties, described for ordinary categories in the books [MP89] of Makkai and Paré and =-=[AR94]-=- of Adàmek and Rosick´y. They were led to discuss categories – now meaning V-categories – with finite limits (in a suitable sense), or more generally with α-small limits, or with filtered colimits (in... |

113 | Metric spaces, generalized logic, and closed categories - Lawvere - 1974 |

107 |
Coherence for compact closed categories
- Kelly, Laplaza
- 1980
(Show Context)
Citation Context ...nabou on closed bicategories. In fact the full result is really present in the one object-closedbicategory situation given by a monoidal closed category, and that was studied in the symmetric case by =-=[KeLa80]-=-. Proposition 6.1 In ANY bicategory, the following statements are equivalent: • (i) the arrow s : A → B has a right adjoint; • (ii) for all g : C → B, the right lifting rlift(s, g) of g through s exis... |

64 |
Accessible categories: the foundations of categorical model theory, volume 104 of Contemporary Mathematics
- Makkai, Paré
- 1989
(Show Context)
Citation Context ...t of Borceux and Quintero [BQ96]. These authors were concerned with extending to the enriched case the notion of accessible category and its properties, described for ordinary categories in the books =-=[MP89]-=- of Makkai and Paré and [AR94] of Adàmek and Rosick´y. They were led to discuss categories – now meaning V-categories – with finite limits (in a suitable sense), or more generally with α-small limits,... |

33 | Variation through enrichment - Betti, Carboni, et al. |

31 |
Structures defined by finite limits in the enriched context 1,” Cahiers de Topologie et Géométrie Différentielle
- Kelly
- 1980
(Show Context)
Citation Context ... op , V] contains the representables K(−, k) and is closed in [K op , V] under Φ-colimits. Example 3.13 6Consider the case when V is locally finitely presentable as a closed category in the sense of =-=[Kel82-2]-=-, and Φ is the class of finite weights as described there; this includes the case where V = Set and Φ is the set of weights for the classical finite colimits. Then Φ ∗ [K] = Φ ∗ (K) is the closure of ... |

31 |
Yoneda structures on 2-categories
- Street, Walters
- 1978
(Show Context)
Citation Context ...-Mod(A, B)(f, [[g, h]]) ∼ = V-Mod(C, B)(fg, h) ∼ = V-Mod(C, A)(g, {| f, h |}). The second isomorphism corresponds by Yoneda to a morphism ǫ : f{| f, h |} → h : C → B which is said, in the language of =-=[StWa78]-=-, to exhibit {| f, h |} as the right lifting of h through f. Such a lifting {| f, h |} is respected by a k : D ◦ �� C when the 2-cell ǫk exhibits {| f, h |}k as the right lifting {| f, hk |} of hk thr... |

16 | Metric spaces, generalized logic, and closed - Lawvere - 1974 |

15 |
Locally presentable and accessible categories,” Cambridge
- Adámek, Rosick´y
- 1994
(Show Context)
Citation Context ...6]. These authors were concerned with extending to the enriched case the notion of accessible category and its properties, described for ordinary categories in the books [MP89] of Makkai and Paré and =-=[AR94]-=- of Adàmek and Rosick´y. They were led to discuss categories – now meaning V-categories – with finite limits (in a suitable sense), or more generally with α-small limits, or with filtered colimits (in... |

11 | Enriched accessible categories
- Borceux, Quintero
- 1996
(Show Context)
Citation Context ...ass of Φ-continuous functors N op → V, where N is a small category admitting Φ-colimits; and we compare these with the Φ-flat functors. For V = Set, some special cases of the results here appeared in =-=[ABLR02]-=-. We have benefited greatly from discussions with Francis Borceux and with Ross Street, both of whom have contributed significantly to the improvement of our exposition; we thankfully acknowledge thei... |

9 |
The closure of a class of colimits
- Albert, Kelly
- 1988
(Show Context)
Citation Context ...hts, which are those ψ for which ψ-limits commute in the base V with colimits having weights in Φ. We show that both these classes are saturated (that is, what was called closed in the terminology of =-=[AK88]-=-). We prove that for the class P of all weights, the classes P + and P − both coincide with the class Q of absolute weights. For any class Φ and any category A, we have the free Φ-cocompletion Φ(A) of... |

7 |
Absolute colimits in enriched categories, Cahiers de Top. et Géom
- Street
- 1983
(Show Context)
Citation Context ... classes is saturated. Section 6 focuses on the class Q = P− where P is the class of all (small) weights; this Q is the class of small projective or atomic weights, which is also, as Street showed in =-=[Str83]-=-, the class of absolute weights. We show that Q is also the class P + of P-flat weights. We recall that a weight φ : Kop → V corresponds to a module φ : I ◦ �� K , while a weight ψ : K → V corresponds... |

7 | Monoidal bicategories and Hopf algebroids, Advance in maths - Day, Street - 1997 |

6 | Cauchy completions - Johnson, Small - 1989 |

5 |
Morita equivalences of enriched categories, Cahiers de Top. et Géom. Diff
- Lindner
- 1974
(Show Context)
Citation Context ...Accordingly, for any A there is a V-category PA having as its objects the accessible presheaves, and with its V-valued hom given by the usual formula ∫ [F a, Ga]; it was first a introduced by Lindner =-=[Lin74]-=-. Any presheaf Aop → V is accessible if A is small, being the left Kan extension of itself along the identity, so that PA coincides with [Aop, V] for a small A. Every representable A(−, b) is accessib... |

3 | Cocompleteness over coverings - Betti - 1985 |

2 |
A theory of enriched sketches, Theory and
- Borceux, Quintero, et al.
- 1998
(Show Context)
Citation Context ... ∼ = P − (B) between subcategories of small projectives. 1 Introduction The present notes had their beginning in the analysis of the results obtained by Borceux, Quintero and Rosicky in their article =-=[BQR98]-=-. These authors were concerned with extending to the enriched case the notion of accessibility and its properties, described for ordinary categories in the book [MP89] of Makkai and Paré and [AR94] of... |

1 |
A theory of enriched sketches, Theory and Applications of Categories 4
- Borceux, Quintero, et al.
- 1998
(Show Context)
Citation Context ...eights and their relation to the Φ-flat weights. 1 Introduction The present observations had their beginnings in an analysis of the results obtained by Borceux, Quintero and Rosick´y in their article =-=[BQR98]-=-, which in turn followed on from that of Borceux and Quintero [BQ96]. These authors were concerned with extending to the enriched case the notion of accessible category and its properties, described f... |

1 |
Lack and Jiri Rosicky, A classification of accessible categories
- Adamek, Borceux, et al.
- 2002
(Show Context)
Citation Context ... generalisations of accessibility as such, but the study of categories with colimits (or limits) of some class, brings considerable simplifications. A recent work by Adamek, Borceux, Lack and Rosicky =-=[ABLR02]-=- also treats a general notion of accessibility parameterised by pair of families of ordinary limits and colimits that commute in Set. Rather than families of ordinary -or conical limits- we consider i... |