Spaces of nonexpanding maps:  categorical properties

V.  Levyts'ka; M. Zarichnyi

The contravariant functor of the spaces of nonexpanding maps into a given bounded metric space acts in the category of metric spaces and nonexpanding maps. We consider the problem of extension of this functor onto the Kleisli categories of some monads.

1. H. Appelgate, Acyclic models and resolvent functors, PhD thesis (Columbia University), 1965.

2. M. Arbib, E. Manes, Fuzzy machines in a category, Bull. Austral. Math. Soc. 13 (1975), no. 2, 169–210.

3. P. T. Johnstone, Adjoint lifting theorems for categories of algebras, Bull. London Math. Soc. 7 (1975), 294–297.

4. M. Barr, Ch. Wells, Toposes, triples and theories, Berlin: Springer-Verlag, 1985.

5. O. Wyler, Algebraic theories of continuous lattices, Lect. Notes in Math. 871 (1981), 390–413.

6. V. Levyts'ka, On extension of the contravariant functor $C_p$ onto categories of multivalued maps, Visn. L'viv. un-tu. Ser. mekh.-mat. 51 (1998), 22--26.

7. J. Vinárek, On extensions of functors to the Kleisli category, Comment. Math. Univ. Carolinae. 18 (1977), 319-327.

8. V. Levyts'ka, On extension of contravariant functors onto the Kleisli category, Matem. Studii. 9 (1998), 125–129.

9. A. Teleiko, M. Zarichnyi, Categorical topology of compact Hausdorff spaces, Lviv: VNTL, 1999.

10. B. Jacobs , E. Poll A Monad for Basic Java Semantics, Lecture Notes in Computer Science, 1816 (2000), 150–164.

11. P. Cenciarelli An Algebraic View of Program Composition, Lecture Notes in Computer Science, 1548 (1998), 325–340.

12. P. S. Mulry, Lifting theorems for Kleisli categories. In: Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science, Volume 802, 1994, 304–319.

13. R. Mirzakhanyan, The infinite iteration of the hyperspace functor. Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1988), no.6, 14–17, 93 (in Russian); translation in Moscow Univ. Math. Bull. bf 43 (1988), no. 6, 15–19.

14. O. R. Nykyforchyn, Probability measures, measurable maps, and convexity: categorical properties, PhD Thesis, Lviv University, 1996. (in Ukrainian)

15. T. N. Radul, The monad of inclusion hyperspaces and its algebras, Ukrain. Mat. Zh. 42 (1990), no. 6, 806–811 (in Russian) translation in Ukrainian Math. J. 42 (1990), no. 6, 712–716.

16. M. Sobral, Contravariant hom-functors and monadicity, Category theory at work (Bremen, 1990), 307–319, Res. Exp. Math., 18, Heldermann, Berlin, 1991.

17. S. Salbany, On Eilenberg-Moore categories induced by chains, Topology Proceedings, 23 (1998), 323–348.

18. V. V. Fedorchuk, Triples of infinite iterations of metrizable functors, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 2, 396–417 (Russian); translation in Math. USSR-Izv. 36 (1991), no. 2, 411–433.

19. A. Bucalo, C. Fohrmann, A. Simpson, Equational lifting monads. CTCS '99: Conference on Category Theory and Computer Science (Edinburgh), Paper No. 29004, 32 pp. (electronic), Electron. Notes Theor. Comput. Sci., 29, Elsevier, Amsterdam, 1999.

	Spaces of nonexpanding maps: categorical properties
Author	V. Levyts'ka, M. Zarichnyi mzar@litech.lviv.ua Institute of Applied,Problems of Mechanics and Mathematics, 3b Naukova Str., Lviv, Ukraine, Faculty of Mechanics and Mathematics, Lviv National University, Universytetska 1, 79000 Lviv, Ukraine
Abstract	The contravariant functor of the spaces of nonexpanding maps into a given bounded metric space acts in the category of metric spaces and nonexpanding maps. We consider the problem of extension of this functor onto the Kleisli categories of some monads.
Keywords	contravariant functor, nonexpanding maps, bounded metric space, metric spaces, Kleisli categories, monads
DOI	doi:10.30970/ms.16.1.3-12
Reference	1. H. Appelgate, Acyclic models and resolvent functors, PhD thesis (Columbia University), 1965. 2. M. Arbib, E. Manes, Fuzzy machines in a category, Bull. Austral. Math. Soc. 13 (1975), no. 2, 169–210. 3. P. T. Johnstone, Adjoint lifting theorems for categories of algebras, Bull. London Math. Soc. 7 (1975), 294–297. 4. M. Barr, Ch. Wells, Toposes, triples and theories, Berlin: Springer-Verlag, 1985. 5. O. Wyler, Algebraic theories of continuous lattices, Lect. Notes in Math. 871 (1981), 390–413. 6. V. Levyts'ka, On extension of the contravariant functor $C_p$ onto categories of multivalued maps, Visn. L'viv. un-tu. Ser. mekh.-mat. 51 (1998), 22--26. 7. J. Vinárek, On extensions of functors to the Kleisli category, Comment. Math. Univ. Carolinae. 18 (1977), 319-327. 8. V. Levyts'ka, On extension of contravariant functors onto the Kleisli category, Matem. Studii. 9 (1998), 125–129. 9. A. Teleiko, M. Zarichnyi, Categorical topology of compact Hausdorff spaces, Lviv: VNTL, 1999. 10. B. Jacobs , E. Poll A Monad for Basic Java Semantics, Lecture Notes in Computer Science, 1816 (2000), 150–164. 11. P. Cenciarelli An Algebraic View of Program Composition, Lecture Notes in Computer Science, 1548 (1998), 325–340. 12. P. S. Mulry, Lifting theorems for Kleisli categories. In: Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science, Volume 802, 1994, 304–319. 13. R. Mirzakhanyan, The infinite iteration of the hyperspace functor. Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1988), no.6, 14–17, 93 (in Russian); translation in Moscow Univ. Math. Bull. bf 43 (1988), no. 6, 15–19. 14. O. R. Nykyforchyn, Probability measures, measurable maps, and convexity: categorical properties, PhD Thesis, Lviv University, 1996. (in Ukrainian) 15. T. N. Radul, The monad of inclusion hyperspaces and its algebras, Ukrain. Mat. Zh. 42 (1990), no. 6, 806–811 (in Russian) translation in Ukrainian Math. J. 42 (1990), no. 6, 712–716. 16. M. Sobral, Contravariant hom-functors and monadicity, Category theory at work (Bremen, 1990), 307–319, Res. Exp. Math., 18, Heldermann, Berlin, 1991. 17. S. Salbany, On Eilenberg-Moore categories induced by chains, Topology Proceedings, 23 (1998), 323–348. 18. V. V. Fedorchuk, Triples of infinite iterations of metrizable functors, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 2, 396–417 (Russian); translation in Math. USSR-Izv. 36 (1991), no. 2, 411–433. 19. A. Bucalo, C. Fohrmann, A. Simpson, Equational lifting monads. CTCS '99: Conference on Category Theory and Computer Science (Edinburgh), Paper No. 29004, 32 pp. (electronic), Electron. Notes Theor. Comput. Sci., 29, Elsevier, Amsterdam, 1999.
Pages	3-12
Volume	16
Issue	1
Year	2001
Journal	Matematychni Studii
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