Self-Duality, Minimal Invariant Objects and Karoubi Invariance in Information Categories

We introduce IP-categories by enriching the notion of I-category (information category) such that every inclusion morphism has a right adjoint or projection. Categories of information systems for various domains in semantics are all examples of IP-categories. We show. that a weak notion of substructure relation between two Scott information systems induces general adjunctions between them. An IP-category with a zero object has the self-dual property that its opposite is again an IP-category. In a complete IP-category with a zero object, limits and colimits of chains of projections and inclusions coincide. As a consequence of the self-duality, a simple characterisation of minimal invariant objects of contravariant and mixed functors on IP-categories is obtained. IP-categories are closed under taking the Karoubi envelope. We use the arrow category of an effectively given IP-category to solve domain equations in categories of continuous information systems effectively.

[1]  Abbas Edalat,et al.  Compact Metric Information Systems (Extended Abstract) , 1992, REX Workshop.

[2]  Steven Vickers,et al.  Applications of Categories in Computer Science: Geometric theories and databases , 1992 .

[3]  Abbas Edalat.  Continuous I-Categories , 1992, LFCS.

[4]  Glynn Winskel,et al.  Using Information Systems to Solve Recursive Domain Equations Effectively , 1984, Semantics of Data Types.

[5]  P. Freyd.  Algebraically complete categories , 1991 .

[6]  Dana S. Scott,et al.  Some Domain Theory and Denotational Semantics in Coq , 2009, TPHOLs.

[7]  Anil Nerode,et al.  Logical Foundations of Computer Science, International Symposium, LFCS 2009, Deerfield Beach, FL, USA, January 3-6, 2009. Proceedings , 1994, LFCS.

[8]  Guo-Qiang Zhang.  DI-Domains as Information Systems* , 1989 .

[9]  A. Jung,et al.  Cartesian closed categories of domains , 1989 .

[10]  M. Fourman,et al.  I-categories and duality , 1992 .

[11]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[12]  S. Lane.  Categories for the Working Mathematician , 1971 .

[13]  Pierre-Louis Curien,et al.  Sequential Algorithms on Concrete Data Structures , 1982, Theor. Comput. Sci..

[14]  Abbas Edalat,et al.  I-Categories as a Framework for Solving Domain Equations , 1993, Theor. Comput. Sci..

[15]  Gordon Plotkin,et al.  Semantics of Data Types , 1984, Lecture Notes in Computer Science.

[16]  Abbas Edalat,et al.  Categories of Information Systems , 1991, Category Theory and Computer Science.

[17]  Samson Abramsky,et al.  Domain Theory in Logical Form , 1991, LICS.

[18]  Gordon D. Plotkin,et al.  The category-theoretic solution of recursive domain equations , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[19]  P. J. Freyd.  Applications of Categories in Computer Science: Remarks on algebraically compact categories , 1992 .