## Relational Databases and Indexed Categories (1992)

Venue: | In Proceedings of the International Category Theory Meeting 1991, CMS Conference Proceedings 13, 391--407, American Mathematical Society |

Citations: | 5 - 2 self |

### BibTeX

@INPROCEEDINGS{Rosebrugh92relationaldatabases,

author = {Robert Rosebrugh and R. J. Wood},

title = {Relational Databases and Indexed Categories},

booktitle = {In Proceedings of the International Category Theory Meeting 1991, CMS Conference Proceedings 13, 391--407, American Mathematical Society},

year = {1992},

pages = {391--407},

publisher = {American Mathematical Society}

}

### OpenURL

### Abstract

. A description of relational databases in categorical terminology given here has as intended application the study of database dynamics, in particular we view (i) updates as database objects in a suitable category indexed by a topos; (ii) L-fuzzy databases as database objects in sheaves. Indexed categories are constructed to model the databases on a fixed family of domains and also all databases for a varying family of domains. Further, we show that the process of constructing the relational completion of a relational database is a monad in a 2-category of functors. Introduction We use the term relation for a subobject of a finite product of objects in a category. Following the relational database literature, we use the term domain for an object of the ambient category (and warn readers that these are not the ordered objects which go by the name "domain" elsewhere in theoretical Computer Science.) A relational database, as defined by E. F. Codd [3], is first of all a family of rela...

### Citations

450 |
The Theory of Relational Databases
- Maier
- 1983
(Show Context)
Citation Context ...hem namely two boolean operations, cartesian product, projection and selection. We remark that the other common operations, with the exception of difference, may be derived from those mentioned below =-=[11]-=-. We are unable to handle the binary difference of relations operation in our setup. Unlike the other operations of relational algebra we use here, it is not monotonic. This spoils functoriality of re... |

333 |
A Relational Model for Large Shared Data Banks
- Codd
- 1970
(Show Context)
Citation Context ...the ambient category (and warn readers that these are not the ordered objects which go by the name "domain" elsewhere in theoretical Computer Science.) A relational database, as defined by E=-=. F. Codd [3]-=-, is first of all a family of relations (or tables) on a family of domains. A heavily used example of domain is the set of character strings over an alphabet. Thus domains should be logically permitte... |

194 |
Topos Theory
- Johnstone
- 1977
(Show Context)
Citation Context ...The point of the construction given is that it can be carried out for S pf for any topos S with NNO. In that case, a + 1 : A 1 + I 1 \Gamma! I 1 above is replaced with the partial morphism classifier =-=[8] for a 1 i-=-n S=I 1 . An update of a database which does not change the current family of relations is one or several of three possible actions on elements ( called "tuples") of the current relations: (... |

130 |
The Formal Theory of Monads
- Street
- 1972
(Show Context)
Citation Context ... fibration [5]. We can now proceed to describe the relational completion monad. We recall that a monad in a 2-category is a 1-cell with a monoid structure given by "unit" and "multiplic=-=ation" 2-cells [17]-=-. Thus in cat 2 a monad is an arrow of cat 2 together with unit and multiplication 2-cells. We note that a monad on A : A 0 \Gamma! A 1 in cat 2 is specified by a monad M 0 in cat on A 0 , a monad M 1... |

75 | Fuzzy functional dependencies and lossless join decomposition of fuzzy relational database systems
- Raju, Majumdar
- 1988
(Show Context)
Citation Context .... It is worth emphasizing that, with the definition we give here, an L-fuzzy database has not only fuzzy relations, but also fuzzy domains and fuzzy index objects for its family of relations. Compare =-=[14]-=-. 3. More indexed categories of databases We have concentrated on a fixed (I-indexed) object A in the discussion above. It is clear that variation of A within A I , and further, variation of I, will d... |

38 |
Categorical semantics for higher order polymorphic lambda calculus
- Seely
- 1986
(Show Context)
Citation Context ...ategories as studied by Par'e and Schumacher [13]. Indexed categories are a widely used categorical tool, but have only begun to be explicitly used in theoretical computer science relatively recently =-=[6,7,12,16]. The &quo-=-t;relational algebra" of relational database theory involves operations which are set-theoretic and other operations which can be defined by a language involving only constants, variables of doma... |

23 | A category-theoretic account of program modules
- Moggi
- 1989
(Show Context)
Citation Context ...ategories as studied by Par'e and Schumacher [13]. Indexed categories are a widely used categorical tool, but have only begun to be explicitly used in theoretical computer science relatively recently =-=[6,7,12,16]. The &quo-=-t;relational algebra" of relational database theory involves operations which are set-theoretic and other operations which can be defined by a language involving only constants, variables of doma... |

6 |
An Introduction to Database Systems,” Fifth Edition
- Date
- 1990
(Show Context)
Citation Context ... portable conceptual structure and are of decreasing interest. Moreover, there is active current research on enhancements and extensions of the relational model. Current editions of the texts by Date =-=[4]-=- or Ullman [18] contain pointers to this work. Research partially supported by grants from NSERC Canada. Diagrams typeset using Catmac. This paper is in final form and no verison of it will be submitt... |

2 |
Fuzzy set theory and topos theory
- Barr
- 1986
(Show Context)
Citation Context ...ce, or more generally a locale. All base categories S above are of this form. Categories of fuzzy sets, after appropriate completion for fuzzy equality, are categories of sheaves on a locale, as Barr =-=[1]-=- has explained. Example 5. Let L be a locale. Following Barr, we will denote by L + the locale L with a new bottom element adjoined. Then ( the completion for fuzzy equality of ) L-fuzzy sets is the c... |

1 |
Cartesian bicategories 1
- Carboni, Walters
- 1987
(Show Context)
Citation Context ...ategory of relations. We end this section with two remarks: (i) the results raise the question whether a relationally closed database is a "2category of relations" in the sense of Carboni an=-=d Walters [2]-=- (= the 2-category rel(E) for a regular category E) -- the answer appears to be no; (ii) the setting we have constructed will be useful for the expression of functional dependencies which are importan... |

1 |
Indexed categories for program development. Cahiers de topologie et g'eometrie diff'erentielle cat'egoriques
- Hilken, Rydeheard
- 1991
(Show Context)
Citation Context ...ategories as studied by Par'e and Schumacher [13]. Indexed categories are a widely used categorical tool, but have only begun to be explicitly used in theoretical computer science relatively recently =-=[6,7,12,16]. The &quo-=-t;relational algebra" of relational database theory involves operations which are set-theoretic and other operations which can be defined by a language involving only constants, variables of doma... |

1 |
Categories in Computer Science and Logic, chapter The Theory of Constructions
- Hyland, Pitts
- 1989
(Show Context)
Citation Context |

1 |
Structures algebriques dans les topos elementaires
- LeSaffre
- 1974
(Show Context)
Citation Context ...is background. For an object I of S we denote the free monoid on I by M (I). Henceforth we assume that M (I) exists in S. It is well known that M (I) exists in any topos with a natural numbers object =-=[10,13]-=-. We also need to assume, and do so for the remainder of this paper that A has finite products. This requires that each A I has finite products preserved by the ff . If A is an object of A I , we will... |

1 |
Indexed Categories and their Applications, chapter Abstract Families and the Adjoint Functor Theorems
- Par'e, Schumacher
- 1978
(Show Context)
Citation Context ...form and no verison of it will be submitted for publication elsewhere. 2 R. ROSEBRUGH & R. J. WOOD The theory of families we use is the theory of indexed categories as studied by Par'e and Schumacher =-=[13]. Indexed -=-categories are a widely used categorical tool, but have only begun to be explicitly used in theoretical computer science relatively recently [6,7,12,16]. The "relational algebra" of relation... |

1 |
On defining objects by recursion in a topos
- Rosebrugh
- 1981
(Show Context)
Citation Context ...this section by finding sufficient conditions for the existence of P 0 (A): Under suitable hypotheses, the required family of finite products can be constructed as a solution to a "recursion prob=-=lem" [15] for the i-=-ndexed functor "crossing with A". We recall that, for S-indexed categories A and B, an indexed functor F : A \Gamma! B is a family of functors F I : A I \Gamma! B I , one for each I in S. Fu... |

1 |
Principles of database and knowledge base systems
- Ullmann
- 1988
(Show Context)
Citation Context ...eptual structure and are of decreasing interest. Moreover, there is active current research on enhancements and extensions of the relational model. Current editions of the texts by Date [4] or Ullman =-=[18]-=- contain pointers to this work. Research partially supported by grants from NSERC Canada. Diagrams typeset using Catmac. This paper is in final form and no verison of it will be submitted for publicat... |