Lawvere completeness in topology. (English) Zbl 1173.18001
Authors’ abstract: It is known since 1973 that Lawvere’s notion of Cauchy-complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper, we introduce the corresponding notion of Lawvere completeness for \((\mathbb{T},{\mathcal V})\)-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones it means weak sobriety while for the latter it means Cauchy completeness. Further, we show that \({\mathcal V}\) has a canonical \((\mathbb{T},{\mathcal V})\)-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; this structure permits us to define a Yoneda embedding in the realm of \((\mathbb{T},{\mathcal V})\)-categories.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18A05 | Definitions and generalizations in theory of categories |
| 18D20 | Enriched categories (over closed or monoidal categories) |
References:
| [1] | Barr, M.: Relational algebras. Lecture Notes in Math. 137, 39–55 (1970) · Zbl 0204.33202 · doi:10.1007/BFb0060439 |
| [2] | Bénabou, J.: Distributors at work. Lecture notes written by Thomas Streicher. www.mathematik.tu-darmstadt.de/treicher/ |
| [3] | Betti, R., Carboni, A., Street, R., Walters, R.: Variation through enrichment. J. Pure Appl. Algebra 29, 109–127 (1983) · Zbl 0571.18004 · doi:10.1016/0022-4049(83)90100-7 |
| [4] | Bonsangue, M.M., van Breugel, F., Rutten, J.J.M.M.: Generalized metric spaces: completion, topology, and powerdomains via the Yoneda embedding. Theoret. Comput. Sci. 193, 1–51 (1998) · Zbl 0997.54042 · doi:10.1016/S0304-3975(97)00042-X |
| [5] | Borceux, F.: Handbook of Categorical Algebra. Vol. 1. Cambridge University Press, Cambridge (1994) · Zbl 0843.18001 |
| [6] | Borceux, F., Dejean, D.: Cauchy completion in category theory. Cahiers Topologie Géom. Differentielle Catég. 27, 133–146 (1986) · Zbl 0595.18001 |
| [7] | Clementino, M.M., Hofmann, D.: Triquotient maps via ultrafilter convergence. Proc. Amer. Math. Soc. 130, 3423–3431 (2002) · Zbl 1008.54011 · doi:10.1090/S0002-9939-02-06472-9 |
| [8] | Clementino, M.M., Hofmann, D.: On limit stability of special classes of continuous maps. Topology Appl. 125, 471–488 (2002) · Zbl 1020.54011 · doi:10.1016/S0166-8641(01)00293-0 |
| [9] | Clementino, M.M., Hofmann, D.: Topological features of lax algebras. Appl. Categ. Structures 11, 267–286 (2003) · doi:10.1023/A:1024274315778 |
| [10] | Clementino, M.M., Hofmann, D.: On extensions of lax monads. Theory Appl. Categ. 13, 41–60 (2004) · Zbl 1085.18006 |
| [11] | Clementino, M.M., Hofmann, D.: Effective descent morphisms in categories of lax algebras. Appl. Categ. Structures 12, 413–425 (2004) · Zbl 1078.18008 · doi:10.1023/B:APCS.0000049310.37773.fa |
| [12] | Clementino, M.M., Hofmann, D.: Exponentiation in V-categories. Topology Appl. 153, 3113–3128 (2006) · Zbl 1155.18007 · doi:10.1016/j.topol.2005.01.038 |
| [13] | Clementino, M.M., Hofmann, D.: Relative injectivity as cocompleteness for a class of distributors. Preprint 08-34, Department of Mathematics, University of Coimbra (2008) · Zbl 1161.18300 |
| [14] | Clementino, M.M., Hofmann, D., Tholen, W.: The convergence approach to exponentiable maps. Portugal Math. 60, 139–160 (2003) · Zbl 1078.54011 |
| [15] | Clementino, M.M., Hofmann, D., Tholen, W.: Exponentiability in categories of lax algebras. Theory Appl. Categ. 11, 337–352 (2003) · Zbl 1050.18006 · doi:10.1023/A:1024458931031 |
| [16] | Clementino, M.M., Hofmann, D., Tholen, W.: One setting for all: metric, topology, uniformity, approach structure. Appl. Categ. Structures 12, 127–154 (2004) · Zbl 1051.18005 · doi:10.1023/B:APCS.0000018144.87456.10 |
| [17] | Clementino, M.M., Tholen, W.: Metric, topology and multicategory–a common approach. J. Pure Appl. Algebra 179, 13–47 (2003) · Zbl 1015.18004 · doi:10.1016/S0022-4049(02)00246-3 |
| [18] | Eilenberg, S., Kelly, G.M.: Closed categories. In: Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pp. 421–562. Springer, New York (1966) · Zbl 0192.10604 |
| [19] | Fletcher, P., Lindgren, W.: A construction of the pair-completion of a quasi-uniform space. Canad. Math. Bull. 21, 53–59 (1978) · Zbl 0393.54019 · doi:10.4153/CMB-1978-009-2 |
| [20] | Fletcher, P., Lindgren, W.: Quasi-uniform Spaces. Lecture Notes in Pure and Applied Maths. Marcel Dekker, New York (1982) |
| [21] | Freyd, P., Scedrov, A.: Categories, Allegories. vol. 39 of North-Holland Mathematical Library. North-Holland, Amsterdam (1990) · Zbl 0698.18002 |
| [22] | Hofmann, D.: An algebraic description of regular epimorphisms in topology. J. Pure Appl. Algebra 199, 71–86 (2005) · Zbl 1072.18002 · doi:10.1016/j.jpaa.2004.12.039 |
| [23] | Hofmann, D.: Exponentiation for unitary structures. Topology Appl. 153, 3180–3202 (2006) · Zbl 1105.18002 · doi:10.1016/j.topol.2005.01.037 |
| [24] | Hofmann, D.: Topological theories and closed objects. Adv. Math. 215, 789–824 (2007) · Zbl 1127.18001 · doi:10.1016/j.aim.2007.04.013 |
| [25] | Hofmann, D.: Injective spaces via adjunction, Technical report. University of Aveiro. arXiv:math.CT/0804.0326. · Zbl 1228.18001 |
| [26] | Hofmann, D., Tholen, W.: Lawvere completion and separation via closure, Technical report, University of Aveiro 2008. arXiv:math.CT70801.0199 · Zbl 1207.18006 |
| [27] | Kelly, G.M.: Basic Concepts of Enriched Category Theory, vol. 64 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1982) · Zbl 0478.18005 |
| [28] | Lawvere, F.W.: Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43, 135–166 (1973); Reprints in Theory and Applications of Categories 1, 1–37 (2002) · Zbl 0335.18006 · doi:10.1007/BF02924844 |
| [29] | Lowen, R.: Approach Spaces, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1997) |
| [30] | Mahmoudi, M., Schubert, C., Tholen, W.: Universality of coproducts in categories of lax algebras. Appl. Categ. Structures 14, 243–249 (2006) · Zbl 1103.18007 · doi:10.1007/s10485-006-9019-6 |
| [31] | Manes, E.: Taut monads and T0-spaces. Theoret. Comput. Sci. 275, 79–109 (2002) · Zbl 1033.18001 · doi:10.1016/S0304-3975(00)00415-1 |
| [32] | Moerdijk, I.: Monads on tensor categories. J. Pure Appl. Algebra 168(2–3), 189–208 (2002) · Zbl 0996.18005 · doi:10.1016/S0022-4049(01)00096-2 |
| [33] | Van Olmen, C.: A study of the interaction between frame theory and approach theory. PhD thesis, University of Antwerp (2005) |
| [34] | Seal, G.J.: Canonical and op-canonical lax algebras. Theory Appl. Categ. 14, 221–243 (2005) · Zbl 1088.18006 |
| [35] | Tholen, W.: Lectures on Lax-Algebraic Methods in General Topology, Lecture 1: \(\mathsf{V}\) -categories, \(\mathsf{V}\) -modules, Lawvere completeness, Haute-Bodeux (2007), http://www.math.yorku.ca/holen/HB07Tholen1.pdf |
| [36] | Wood, R.J.: Abstract proarrows. I. Cahiers Topologie Géom. Differentielle Catég. 23(3), 279–290 (1982) · Zbl 0497.18012 |
| [37] | Wood, R.J.: Ordered sets via adjunctions. In: Categorical Foundations. Encyclopedia Math. Appl., vol. 97, pp. 5–47. Cambridge University Press, Cambridge (2004) · Zbl 1044.06001 |
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