This book presents the formal definition of fundamental transformations in Category Theory as a mathematical language to be used in Computer Science modelling. The book focuses in particular on models with Global and Internal symmetries (in analogy to Field Theories like Quantum Mechanics and General Relativity). The second part of the book is dedicated to more advanced applications of Category Theory to Computer Science.
Author(s): Zoran Majkić
Publisher: De Gruyter
Year: 2023
Language: English
Pages: 436
1 Basic transformations of categories: Hierarchy of n-dimensional levels ? 1
1.1 The n-dimensional levels: A nonset-based definition of natural numbers ? 1
1.2 Comma lifting for n-dimensional levels ? 6
1.3 Introduction to comma-induction in the n-dimensional hierarchy ? 16
1.4 Comma-induction of the adjunctions ? 20
2 Comma-propagation transformations: Global categorial symmetries ? 29
2.1 Introduction to general comma-propagation ? 29
2.2 Comma-propagation of functors and natural transformations ? 34
2.3 Comma-propagation of (co)limits ? 40
2.4 Example: Comma-propagation and infinite hierarchy of small-complete
categories ? 50
2.5 An analogy between physical and abstract categorial Global symmetries:
Adjunctions-as-fields ? 57
2.5.1 Analogy between adjunctions and metric tensor field with Einstein–Hilbert
action ? 60
2.5.2 Comma-propagation transformation symmetries ? 65
3 Arrows-to-objects conceptual transformation: Internal categorial
symmetry ? 71
3.1 Introduction to categorial internal symmetry of primitive categorial
concepts ? 71
3.2 Internal symmetry and metacategory ? 77
3.3 Conceptually closed categories: A topology ? 83
3.4 Symmetry-extended categories ? 94
3.5 Symmetry hierarchy upper bound: Imploded categories ? 101
4 Internal symmetry and logical deduction ? 111
4.1 Natural deduction system ? 111
4.1.1 First solution for tagging techniques of natural deduction ? 111
XXX ? Contents
4.1.2 Sequent-based solution for tagging techniques ? 116
4.2 Definition of symmetry-extended category ND for natural deduction ? 119
4.2.1 The properties of the covariant implication functor ? 125
4.2.2 Local Cartesian Closed Adjunction (pCCC) ? 128
4.2.3 The n-dimensional levels of natural deduction categories ? 133
4.3 Symmetry-extended category IC for propositional intuitionistic
calculus ? 138
5 Internal symmetry and lambda calculus ? 144
5.1 Introduction to lambda calculus ? 144
5.2 Reflexive objects in the Cartesian closed categories ? 149
5.3 Conceptually-closed CCC and its subcategories of idempotents ? 156
5.4 Internal categorial symmetry and fixed-point operators ? 166
5.5 Topological K-theory, idempotent completion and internal categorial
symmetry ? 169
6 Internal symmetry and theory of processes: Strong bisimulation of
computation trees ? 173
6.1 Introduction to transition systems and their bisimulations ? 173
6.2 Regular languages, automata and internal categorial symmetry ? 177
6.3 Symmetry-extended category of finite labeled trees ? 183
6.4 Internal symmetry of the category of relations ? 191
6.5 Transition systems as fixed points in the n-dimensional level Rel3 ? 196
6.6 Strong bisimulations as fixed points in the n-dimensional level Rel3 ? 201
6.6.1 Mathematics via symmetry: Reduction of many valued into 2-valued
logic ? 207
6.6.2 Many-valued knowledge invariance through modal logic transformations:
Semantic reflection ? 212
7 Internal symmetry and data integration theory ? 220
7.1 DB (Database) category ? 220
7.1.1 Morphism properties of DB category ? 235
7.1.2 Power-view endofunctor and monad T ? 249
7.1.3 Duality ? 254
7.2 Objects of DB: Basic operations and equivalence relations ? 257
7.2.1 Data federation operator in DB ? 257
7.2.2 Data separation operator in DB ? 258
7.2.3 The (strong) behavioral equivalence for databases ? 261
7.2.4 Weak observational equivalence for databases ? 263
7.3 Internal categorial symmetry of DB category ? 266
7.3.1 (Co)products ? 270
7.3.2 (Co)Limits and exponentiation ? 274
Contents ? XXXI
7.3.3 Kleisli semantics for database mappings ? 282
7.4 Partial ordering for databases ? 288
7.4.1 Matching tensor product ? 292
7.4.2 Merging operator ? 295
7.4.3 Universal algebra considerations ? 298
7.4.4 Algebraic database lattice ? 302
7.5 Enrichment ? 313
7.5.1 DB is a V-category enriched over itself ? 315
7.5.2 Internalized Yoneda embedding ? 320
A Appendix ? 323
A.1 Introduction to lattices, algebras and logics ? 323
A.1.1 Introduction to deductive logic and binary sequent calculus ? 327
A.1.2 Introduction to first-order logic and Tarski’s interpretations ? 329
A.1.3 Introduction to multimodal logics and Kripke semantics ? 333
A.2 Basic category theory ? 334
A.3 Introduction to RDB, database mappings and DB category ? 348
A.3.1 Basic database concepts ? 351
A.3.2 Database observations: Idempotent power-view operator ? 357
A.3.3 Logic versus algebras: Categorification by operads ? 360
A.3.4 Sketches and functors into the DB category ? 363
A.3.5 Semantics of DB schema mappings: Information fluxes ? 370
A.4 Introduction to field theory and symmetries ? 381
A.4.1 Vector fields on curved differentiable manifolds ? 388
A.4.2 Transformation of coordinates ? 392
Bibliography ? 395
Index ? 403
