Restriction categories II: Partial map classification. (English) Zbl 1023.18005
This paper is a sequel to the same authors’ [J. R. B. Cockett and S. Lack, ibid. 270, 223-259 (2002; Zbl 0988.18003)]. There they introduced restriction categories for working with abstact categories of partial maps, and showed an representation theorem justifying their regarding restriction categories as abstract categories of partial maps. This paper considers when a monad has a Kleisli category which is abstractly a classified category of partial maps. In the presence of products the question has been answered by A. Bucalo, C. Führmann and A. Simpson [ibid. 294, 31-60 (2003; Zbl 1022.18003)]. Moving from the \(p\)-category setting to the restriction category setting, the authors answer the same question without assuming the presence of products.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18C20 | Eilenberg-Moore and Kleisli constructions for monads |
| 68Q99 | Theory of computing |
| 18B99 | Special categories |
| 18D99 | Categorical structures |
References:
| [1] | J. Bénabou, Introduction to bicategories, Reports of the Midwest Category Seminar, Lecture Notes in Mathematics, Vol. 106, 1967, pp. 1-77.; J. Bénabou, Introduction to bicategories, Reports of the Midwest Category Seminar, Lecture Notes in Mathematics, Vol. 106, 1967, pp. 1-77. · Zbl 1375.18001 |
| [2] | A. Bucalo, C. Führmann, A. Simpson, An equational notion of lifting monads, Theoret. Comput. Sci. this Vol. (2003).; A. Bucalo, C. Führmann, A. Simpson, An equational notion of lifting monads, Theoret. Comput. Sci. this Vol. (2003). |
| [3] | Carboni, A., Bicategories of partial maps, Cah. de Top. Geom. Differential, 28, 111-126 (1987) · Zbl 0631.18002 |
| [4] | J.R.B. Cockett, S. Lack, Restriction categories I: categories of partial maps, Theoret. Comput. Sci. 270 (2001) 223-259. Available electronically from; J.R.B. Cockett, S. Lack, Restriction categories I: categories of partial maps, Theoret. Comput. Sci. 270 (2001) 223-259. Available electronically from · Zbl 0988.18003 |
| [5] | J.R.B. Cockett, S. Lack, Restriction categories III: partial structures, in preparation.; J.R.B. Cockett, S. Lack, Restriction categories III: partial structures, in preparation. · Zbl 1123.18003 |
| [6] | J.R.B. Cockett, D. Spencer, Strong categorical datatypes I, Category Theory 1991 (Montreal, PQ, 1991) CMS Conf. Proc. 13, Amer. Math. Soc., Providence RI, 1992, pp. 141-169.; J.R.B. Cockett, D. Spencer, Strong categorical datatypes I, Category Theory 1991 (Montreal, PQ, 1991) CMS Conf. Proc. 13, Amer. Math. Soc., Providence RI, 1992, pp. 141-169. · Zbl 0792.18008 |
| [7] | Di Paola, R. A.; Heller, A., Dominical categories: recursion theory without elements, J. Symbolic Logic, 52, 595-635 (1987) · Zbl 0649.03032 |
| [8] | Führmann, C., Direct Models of the Computational Lambda-Calculus, MFPS XV (= Electron. Notes Theor. Comput. Sci., Vol. 20 (1999), Elsevier: Elsevier Amsterdam · Zbl 0924.68029 |
| [9] | Johnstone, P. T., Topos Theory, L.M.S Monographs, Vol. 10 (1977), Academic Press: Academic Press London · Zbl 0368.18001 |
| [10] | Johnstone, P. T., Variations on the bagdomain theme, Theoret. Comput. Sci., 136, 3-20 (1994) · Zbl 0874.18003 |
| [11] | Kelly, G. M.; Power, A. J., Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, J. Pure Appl. Algebra, 89, 163-179 (1993) · Zbl 0779.18003 |
| [12] | Kock, A., Strong functors and monoidal monads, Arch. Math., 23, 113-120 (1972) · Zbl 0253.18007 |
| [13] | Manes, E. G., Algebraic Theories (1976), Springer: Springer New York · Zbl 0489.18003 |
| [14] | Mulry, P. S., Partial map classifiers and partial cartesian closed categories, Theoret. Comput. Sci., 99, 141-155 (1992) · Zbl 0769.68084 |
| [15] | Mulry, P. S., Monads and algebras in the semantics of partial data types, Theoret. Comput. Sci., 136, 109-123 (1994) · Zbl 0874.68183 |
| [16] | Mulry, P. S., Lifting theorems for Kleisli categories, Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science, Vol. 802 (1994), Springer: Springer Berlin · Zbl 1509.18004 |
| [17] | Robinson, E. P.; Rosolini, G., Categories of partial maps, Inform. and Comput., 79, 94-130 (1988) · Zbl 0656.18001 |
| [18] | G. Rosolini, Continuity and effectiveness in topoi, D.Phil Thesis, Oxford University, 1986.; G. Rosolini, Continuity and effectiveness in topoi, D.Phil Thesis, Oxford University, 1986. |
| [19] | Street, R. H., The formal theory of monads, J. Pure Appl. Algebra, 2, 149-168 (1972) · Zbl 0241.18003 |
| [20] | J.R.B. Cockett, S. Lack, The extensive completion of a distributive category, Theory Appl. Categ. 8 (2001) 541-554. Available electronically from; J.R.B. Cockett, S. Lack, The extensive completion of a distributive category, Theory Appl. Categ. 8 (2001) 541-554. Available electronically from · Zbl 1005.18003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
