Lecture 3: Object Concepts as Categories

1. Lecture 3: Object Concepts as Categories Nick Rossiter, Computing Science, Newcastle University, England B.N.Rossiter@newcastle.ac.uk http://www.cs.ncl.ac.uk/people/b.n.rossiter/

2. Introduction • Look at use of categories for application: • Modelling of object information systems • Introduce further categorical constructions: • Functors • Cartesian Closed Categories • Natural Transformations

3. Classes as Categories • Classes have • identifiers: represented by a categorical object • attributes: represented by an arrow from the identifier (as source) to each attribute (as target) arrows of type assignment arrows can be monic or epic

4. Classes as Categories -- Methods • methods: represented by arrows from one or more attributes to another attribute. arrows of type function • We define the Category Class in the universal category Set to have the above constructions.

5. Example Class • Category Student dependency function StudentId StudentName {Module Module Title Mark} AverageMark Will structure some of this later as pullback (N:M)

6. Example Class - Observations • StudentId is initial object of category • Identifiers/keys are usually initial objects • No terminal object as it stands • StudentId StudentName is not monic nor epic • Module ModuleTitle is monic (1:1)

7. Example Class - Types • The class category Student forms one type for use elsewhere in the system • Arrows are also typed e.g. • Module ModuleTitle is monic • StudentId is in the domain available for student identifiers in the discrete category S: s: SStudentId (or s:1S StudentId) There must be a mapping for the instance s from the discrete category S to StudentId

8. Constraints: Example of Keys • Constraints such as normalization tests can be applied e.g. Boyce-Codd Normal Form (BCNF) • For a relation, say R(a,b,c,d), can determine the candidate keys from the given (non-trivial) functional dependencies. • Can then test whether every determinant (source of non-trivial functional dependency) is a candidate key

9. Procedure for Normalization Test 1 • Take category CLASS • Transform CLASS to POS where: • the objects are the powerset of those in CLASS • the arrows are the orderings by projection (trivial functional dependencies) • Create a category DEP holding the non-trivial functional dependencies. • Embed POS and DEP in a category PRJ

10. Procedure for Normalization Test 2 • Determine limit of singleton attributes in PRJ (greatest lower bound) if it exists • If no limit, determine maximal lower bounds. • Assign limit or maximal lower bounds to candidate key.

11. Example of Normalization Test 1 • CLASS: objects a,b,c, arrow a b identity arrows for each object • POS: objects powerset of a,b,c (23) arrows trivial dependencies as projections

12. Example of Normalization Test 2 • PRJ: as POS but with addition of non-trivial dependency a b empty {a} {b} {c} {a,b} {a,c} {b,c} trivial {a,b,c} non-trivial

13. Example of Normalization Test 3 • Let Y = {{a}, {b}, {c}} (singleton attributes) • Limit Y = {a,c} (greatest lower bound) • Candidate key = {a,c} • Determinant = {a} • So not BCNF as not every determinant is a candidate key.

14. Further Category Constructions: The Functor • Categories are mapped from one to another by a higher-order function called a Functor. • A functor F maps from category A to B by: • assigning each F(a) to B • assigning each F(f) to B where a is an object in A and f is an arrow in A

15. Functors Preserve Composition • F(f1) o F(f2) =F(f1 o f2) F c = F(c) f h F(f) F(h) b a F(b) F(a) g F(g)

16. Functors Preserve Identities • Functors preserve identities: F(1a) = 1F(a) • That is: F (1a : a a) = 1F(a) : F(a) F(a)

17. Functors are Arrows • They are composable F : A B G: B C C H = G o F G H B A F

18. Functors are Associative • Their evaluation is associative. • The order of evaluation of three or more composed functors does not matter

19. Scope of Functors • Can preserve source category structure in target category: F: A B Full and faithful functor: A,B same structure

20. Free Functors • Can assign additional structures • For example assign to each set its powerset (all combinations): POWER: CLASSPOS assigns the powerset of the set, say Y, in CLASS to POS. • Free functor

21. Forgetful Functors • Can forget structure • For example forget set functions in a Set category. REMOVE_FUNCTIONS: Set Discrete • Underlying (Forgetful) functor • Returns elements only

22. Functors for Embedding The functors EMBED1: POSPRJ EMBED2: DEPPRJ embed the arrows of POS and DEP in PRJ. Inclusion functor

23. Identity Functor • Can identify a category 1A : AA 1A is identity functor for category A

24. Relationships between Classes • Association: by pullback Student pil f Student xmark Exam Mark pir g Exam

25. Semantics of Pullbacks • Gives constraints of entity-relationship modelling of relationships: • cardinality of 1:1, 1:N, N:M • mandatory or optional participation • can paste together pullbacks to give n-ary relationships • colimit of diagram includes the link object, the name of which appears as the product suffix

26. Inheritance • Through coproducts: as cone Manager+Employee Manager Employee

27. Inheritance as Commuting Diagram • More detailed representation as commuting diagram f Manager il U Manager+ g ir Employee Employee • Gives unqualified union (U is universal object) il o f = ir o g

28. Cartesian Closed Categories • Widespread use in computing science • A Cartesian closed category has: • a terminal object • products • exponentials (connectivity - no discrete objects)

29. Examples Cartesian Closed • The diagrams for products, coproducts, pullbacks and pushouts are examples of Cartesian closed categories • Further example: A • A X B X C B emptyset C has products, terminal object (empty set) and connectivity (no discrete objects)

30. Object Instantiation • The Extension -- the category OBJECT representing the objects associated with each category • Functors relate intension to extension: Data: CLASS OBJECT • and extension to intension: Name: OBJECT CLASS

31. Constraints with Functors • The functors preserve types, composition, identities and limits of the intension in the extension. P o E = I TYPE E = Name P I CLASS OBJECT E

32. Information System Queries • D : CLASSOBJECT D is database with CLASS as intension and OBJECT as extension • Query q on D returns subset of database DS with intension CLASSDS and extesnion OBJECTDS: • DS : CLASSDS OBJECTDS

33. Query DS • Query is: q: DS D DS is subset of D Both DS and D are functors

34. Natural Transformation • q is natural transformation as it maps from one functor to another • Functors must be of similar type • Assigns underlying categorical objects and arrows of one functor to another

35. Simple View D CLASS OBJECT q CLASSDS OBJECTDS DS

36. Details of Natural Transformation DS(source(f)) qa D(source(f)) DS(f) D(f) DS(target(f)) qb D(target(f)) For each f in CLASSDS, the equation: D(f) o qa = qb o DS(f) must hold

37. Further Reading • Categories for the Working Mathematician Mac Lane, Saunders 2nd edition Springer-Verlag (1998). • Prototyping a Categorical Database in P/FDM, Nelson, D A, & Rossiter, B N, Proc. 2nd Int. Workshop Advances in Databases and Information Systems (ADBIS'95), Moscow, 27-30 June 1995, Springer-Verlag Workshops in Computing, ISBN 3-540-76014-8, 432--456 (1996).