Home

# Restriction categories III: colimits, partial limits and extensivity

Abstract

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about partial maps as they have a purely algebraic formulation.

In this paper we consider colimits and limits in restriction categories. As the notion of restriction category is not self-dual, we should not expect colimits and limits in restriction categories to behave in the same manner. The notion of colimit in the restriction context is quite straightforward, but limits are more delicate. The suitable notion of limit turns out to be a kind of lax limit, satisfying certain extra properties.

Of particular interest is the behaviour of the coproduct, both by itself and with respect to partial products. We explore various conditions under which the coproducts are ‘extensive’ in the sense that the total category (of the related partial map category) becomes an extensive category. When partial limits are present, they become ordinary limits in the total category. Thus, when the coproducts are extensive we obtain as the total category a lextensive category. This provides, in particular, a description of the extensive completion of a distributive category.

References
Hide All
Bucalo, A., Führmann, C. and Simpson, A. (1999) Equational lifting monads. In: CTCS'99: Conference on Category Theory and Computer Science (Edinburgh). Electron. Notes Theor. Comput. Sci. 29 Paper No. 29004.
Carboni, A. (1987) Bicategories of partial maps. Cah. de Top. Geom. Diff. 28 111126.
Carboni, A. (1991) Matrices, relations, and group representations. J. Algebra 136 497529.
Carboni, A., Lack, S. and Walters, R. F. C. (1993) Introduction to extensive and distributive categories. J. Pure Appl. Algebra 84 145158.
Cockett, J. R. B. (1993) Introduction to distributive categories. Mathematical Structures in Computer Science 3 277307.
Cockett, J. R. B. and Lack, S. (2002) Restriction categories I: Categories of partial maps. Theoretical Computer Science 270 223259.
Cockett, J. R. B. and Lack, S. (2003) Restriction categories II: Partial map classification. Theoretical Computer Science 294 61102.
Cockett, J. R. B. and Lack, S. (in preparation) Restriction categories IV: Enriched restriction categories.
Cockett, J. R. B. and Lack, S. (2001) The extensive completion of a distributive category. Theory Appl. Categ. 8 541554.
Curien, P.-L. and Obtulowicz, A. (1989) Partiality, Cartesian closedness, and toposes. Inform. and Comput. 80 5095.
Di Paola, R. A. and Heller, A. (1987) Dominical categories: recursion theory without elements. J. Symbolic Logic 52 595635.
Freyd, P. J. and Scedrov, A. (1990) Categories, allegories, North-Holland Mathematical Library 39
Hoehnke, H.-J. (1977) On partial algebras. Colloq. Math. Soc. János Bolyai 29 373412.
Kock, A. (1972) Strong functors and monoidal monads. Arch. Math. 23 113120.
Mac Lane, S. (1971) Categories for the Working Mathematician, Springer-Verlag.
Robinson, E. P. and Rosolini, G. (1988) Categories of partial maps. Information and Computation 79 94130.
Recommend this journal

Mathematical Structures in Computer Science
• ISSN: 0960-1295
• EISSN: 1469-8072
• URL: /core/journals/mathematical-structures-in-computer-science
Who would you like to send this to? *

×

## Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 23 *