Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-18T06:33:59.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterising E-projectives via Comonads

Published online by Cambridge University Press:  12 May 2015

WENG KIN HO
WENG KIN HO*
Affiliation:
Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 637616Singapore Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper demonstrates the usefulness of a comonadic approach to give previously unknown characterisation of projective objects in certain categories over particular subclasses of epimorphisms. This approach is a simple adaptation of a powerful technique due to Escardó which has been used extensively to characterise injective spaces and locales over various kinds of embeddings, but never previously for projective structures. Using some examples, we advertise the versatility of this approach – in particular, highlighting its advantage over existing methods on characterisation of projectives, which is that the comonadic machinery forces upon us the structural properties of projectives without relying on extraneous characterisations of the underlying object of the coalgebra arising from the comonad.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 4: Symposium on Domain Theory (ISDT 2013) , May 2017 , pp. 491 - 506
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. and Jung, A. (1994). Domain theory. Handbook of Logic in Computer Science, volume 3, Clarendon Press, Oxford.Google Scholar
Balbes, R. (1967). Projective and injective distributive lattices. Pacific Journal of Mathematics 21 405–420.Google Scholar
Balbes, R. and Dwinger, P. (1975). Distributive Lattices, University of Missouri Press.Google Scholar
Banaschewski, B. and Niefeld, S. B. (1991). Projective and supercoherent frames. Journal of Pure and Applied Algebra 70 45–51.Google Scholar
Crown, G. D. (1970). Projectives and injectives in the category of complete lattices with residuated mappings. Mathematische Annalen 187 295–299.Google Scholar
Davey, B. A. and Priestley, H. A. (1990). Introduction to Lattices and Order, Cambridge University Press.Google Scholar
Erné, M. (1999). Z-continuous posets and their topological manifestation. Applied Categorical Structures 7 31–70.Google Scholar
Escardó, M. H. (1997). Injective spaces via the filter monad. Topology Proceedings 22 97–110.Google Scholar
Escardó, M. H. (1998). Properly injective spaces and function spaces. Topology and its Applications 89 (1–2) 75–120.Google Scholar
Escardó, M. H. (2003) Injective locales over perfect embeddings and algebras of the upper powerlocale monad. Applied General Topology 4 (1) 193–200.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. W. and Scott, D. S. (2003). Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications, volume 93, Cambridge University Press, Cambridge.Google Scholar
Ho, W. K. and Zhao, D. (2009). Lattices of Scott-closed sets. Commentationes Mathematicae Universitatis Carolinae 50 (2) 297–314.Google Scholar
Johnstone, P. T. (1982). Stone Spaces, Cambridge Studies in Advanced Mathematics, volume 3, Cambridge University Press.Google Scholar
Kock, A. (1995). Monads for which structures are adjoint to units. Journal of Pure Applied Algebra 104 41–59.CrossRefGoogle Scholar
Mac Lane, S. (1998). Categories for the Working Mathematician, 2nd edition, Springer-Verlag.Google Scholar
Nai, Y. T. (2012). Order Structures in Algebra, Confirmation report for PhD, National Institute of Education, Nanyang Technological University.Google Scholar
Poigné, A. (1992). Basic category theory. Handbook of Logic in Computer Science, volume 1, Oxford University Press, New York.Google Scholar
Scott, D. S. (1972). Continuous lattices. Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics, Springer-Verlag.Google Scholar
Sikorski, R. (1948). A theorem on extensions of homomorphisms. Ann. Soc. Polon. Math. 21 332–335.Google Scholar
Wright, J. B., Wagner, E. G. and Thatcher, J. W. (1978). A uniform approach to inductive posets and inductive closure. Theoretical Computer Science 7 57–77.Google Scholar
Zhao, B. and Wang, K. Y. (2010). Z-Quantales and its Categorical Properties. Acta Mathematica SINICA, Chinese Series 53 (5) 997–1006.Google Scholar
Zhao, D. (1992) Generalisation of Continuous Lattices and Frames, PhD, University of Cambridge.Google Scholar
Zhao, D. (1997). On projective frames. Canadian Mathematical Bulletin 40 (1) 39–46.Google Scholar