1. Categories for Information
Nick Rossiter
1st March 2006
School of Computing, Engineering and Information Sciences
Northumbria University
09/04/2019

2. Information Systems Very diverse Usually multilevel
A stand-alone piece of information
Is valueless
Needs to be typed
Needs to be related
Needs to be placed in context
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3. Example of Context This is Herring Gull argenteus (subspecies)
Trinomial – 3-level name is:
Larus argentatus argenteus
Kingdom: Metazoa ((=Animalia) multicellular animals)
Phylum: Chordata (chordates)
Class: Aves (birds)
Order: Charadriiformes (gulls and shore birds)
Family: Laridae (gulls, terns)
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4. Interoperability An area where context is paramount is
the ability to request and receive services between various systems and use their functionality.
More than data exchange.
Implies a close integration
Various kinds dependent on ambition:
E.g. syntactic, semantic, structural and organisational
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5. Motivation/Problems Linking of Different Systems (Current/legacy)
Homogeneous models
Difficult enough
Different viewpoints in modelling
E.g. library system
A fine could be:
A relational table
A column in a table
A value in an income ledger
Inconsistent use of modelling features
Systems that achieve interoperability in such circumstances are ranked
As semantically interoperable
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6. Motivation/Problems 2 Heterogeneous Models Far more difficult
In addition to different semantic viewpoints
Diverse modelling constructions
Data structures
Objects, relations, records
Process
Business process, procedures, methods
More recent models are semantically richer
More scope for variation in style
Systems that achieve interoperability in such circumstances are ranked
As structurally (or organisationally) interoperable
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7. Demands for Interoperability
Business Needs
Data warehousing
Web Warehousing
GRID
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8. Attempted Solutions RDF (Resource Description Framework)
Triples (uri – e.g. resource/property/statement)
From W3C (XML basis)
MOF/MDA (Meta Object Facility/Model Driven Architecture)
Meta Meta is better-better!
Relates classes in different systems
From OMG (UML basis but claimed to be extensible)
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9. Attempted Solutions 2 Ontologies Defines meaning of data
Being
Defines meaning of data
Like a dictionary
But is usually much more
Everything is defined in context
Multi-level definitions
No clear consensus
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10. Formal Basis For preceding techniques These are partial:
Some set theoretic justifications
These are partial:
Emphasis on a level
Contrived multi-level
Above all – lack concept of naturality
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11. Categories Category Theory Developed from 1940s
Many pure mathematicians
Eilenberg, Mac Lane, Kan, Lawvere, Barr, Wells, Johnstone
Much improved presentation since 1970s
Saunders Mac Lane “Categories for the Working Mathematician” 2nd ed Springer (2000)
Barr & Wells “Category Theory for Computing Science” 3rd ed CRM (1999).
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12. Applied Categories Physics including quantum studies Databases
John Baez
Databases
Bob Rosebrugh, Michael Johnson, Zinovy Diskin, Lellahi & Spyratos
Business process
Arthur ter Hofstede
Computer program semantics
Much work e.g. Cambridge
Programs to Support Category Theory
OCaml (ENRIA, France)
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13. Abstract Nonsense One might ask "Why category theory?“
Category theory is known as highly abstract mathematics.
Some call it abstract nonsense.
It chases abstract arrows and diagrams, proves nothing about those arrows and diagrams, rarely talks about what arrows are for and often concepts go beyond one's imagination.
However, when this 'abstract nonsense' works, it is like magic. One may discover a simple theorem actually means very deep things and some concepts beautifully unify and connect things which are unrelated before.
[Tatsuya Hagino. A Categorical Programming Language. PhD Thesis, University of Edinburgh, 1987]
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14. The Simplest Category Discrete Category Identity arrows (objects) only
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15. A Not Very Useful Category
6 arrows
Not connected
Does not conflict
with axioms
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16. Basic Category Illustrates 2 axioms when connections made.
Composition:
h = g o f
Associativity:
r o (q o p) =
(r o q) o p
Also unit law
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17. Cartesian Closed Category
Basis of much
Computing Science
Research in CT C
P+P+P
Has identity, products,
limits, coproducts
PxPxP
Identity functor
1C: C - C
Initial object PxPxP
provides handle on category
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18. Functors Map from one category to another Preserve composition
E.g. F: C  D
Preserve composition
Various kinds
Identity (map category to itself)
Free (add structure)
Underlying/Forgetful (remove structure)
Adjoint (two-way relationship)
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19. Natural Transformations
Map from one functor to another
E.g. : F  G
Functors must be of same variance
No further levels are needed
Comparison of natural transformations is a natural transformation
E.g. :   
An arrow in a category is defined in context as unique up to natural isomorphism
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20. Informal Requirements for IS Architecture
Concepts
MetaMeta Policy
Meta Organize
Classify Instantiate
Constructs
Schema Types
Named Data Values
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Downward arrows are intension-extension pairs

21. Formalising the Architecture
Requirements:
mappings within levels and across levels
bidirectional mappings
closure at top level
open-ended logic
relationships (product and coproduct)
Choice: Category theory as used in mathematics as a workspace for relating different constructions
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22. Blue – category, red functor, green nat trans
Figure 2: More Detailed Interpretation of Levels
in Category Theory: Natural Schema
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23. Godement Calculus Manipulates categorical diagrams
Is a natural calculus
Provides rules showing:
composition of functors and natural transformations is associative
natural transformations can be composed with each other
Developed by Godement in 1950s
Has Interchange laws
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24. Comparison of Three Systems
Figure 9: Organisational Interoperability
for use with Godement Calculus.
Variable Policy
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25. Equations (Figure 6) for Godement Calculus from Simmons
Equations (6) interchange, (7)-(8) associativity, (9) permutation, (10) different paths (composition)
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26. Technical Conditions for Interoperability
That our categories obey the rules of category theory
every triangle in the diagram commutes (composition)
order of evaluating arrows is immaterial (associativity)
identity arrows are composable with other arrows
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27. Anticipated Problems 1 Type Information
ICEIS 2005
Anticipated Problems 1 Type Information
Semantic annotation needed
To obtain metameta types from implicit sources
Needs open architecture
Agents have potential
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28. Anticipated Problems 2 Composition Failure
Partial functions
Most categories are based on total functions
In real world many mappings are partial
not all of the source objects participate in a relationship (mapping)
Composition breaks down in a ‘total function’ category if a partial function occurs
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29. Summary
Formal four-level architecture promising for tackling interoperability:
Use of category theory in natural role
Structure and relations through arrows (identity, category, functor, natural transformation)
Manipulate through Godement calculus
Problems:
Composition failure (particularly with partial functions)
Need semantic annotation
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30. Prospects – PhD students
Robert Warrender (Sunderland) – testing 4-level ct architecture for relational and o-o databases
Dimitris Sisiaridis (Northumbria)– using 4-level ct architecture for security
Tim Reichert (Heilbronn/Northumbria) – using languages such as Qi for realising interoperability with ct. Development of tool for demonstrating technique.
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31. Recent/Future Publications
Rossiter, Nick, & Heather, Michael, Conditions for Interoperability, 7th International Conference on Enterprise Information Systems (ICEIS), Florida, USA, May 2005, (2005)
Rossiter, Nick, Heather, Michael, & Nelson, David, A Natural Basis for Interoperability, I-ESA’06, Interoperability for Enterprise Software and Applications Conference, University of Bordeaux, March 2006, 12pp, Springer (2006). Also leading to journal paper as part of set-up. EU Athena initiative
Rossiter, Nick, & Heather, Michael, Free and Open Systems Theory, EMCSR-2006, 18th European Meeting on Cybernetics and Systems Research, University of Vienna, April 2006, 6pp (2006).
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