Abstract
Viewing neighbourhood operators as lax natural transformations introduces an efficiency in calculations and proofs and suggests further applications. To highlight the advantages of this approach, classes of open, closed, initial and final morphisms are studied. In addition new proofs are given to previous results and a new example that departs from the current factorisation system paradigm is exhibited.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Castellini, G.: Interior operators in a category: idempotency and heredity. Top. Appl. 158, 2332–2339 (2011)
Castellini, G.: Interior operators, open morphisms and the preservation property. Appl. Categor. Struct. (2013). 10.1007/s10485-013-9337-4
Clementino, M.M., Giuli, E., Tholen, W.: What is a quotient map with respect to a closure operator. Appl. Cat. Struct. 9, 139–151 (2001)
Clementino, M. M., Giuli, E., Tholen, W.: A functional approach to general topology, Categorical Foundations. In: Pedicchio, M.C., Tholen, W. (eds.) 103-163. Encyc. Math. Appl., 97, Cambridge Univ. Press, Cambridge (2004)
Császár, Ȧ.: Foundations of general topology (A translation by Mrs. K. Császár of Fondements de la topologie générale, originally published in French by Akadémiai Kiadó (1960)), 380. Pergamon Press, Oxford (1963)
Dikranjan, D., Tholen, W.: Categorical structure of closure operators. Kluw. Acad. Pub. 356 (1996). Dordrecht
Dikranjan, D., Tholen, W.: Dual closure operators and their applications. Top. Appl. 439, 373–416 (2015)
Engelking, R.: General Topology, vol. 529. Heldermann Verlag, Germany (1989)
Giuli, E., Šlapal, J.: Neighbourhoods with respect to a categorical closure operators. Acta. Math. Hungar. 124(1-2), 1–14 (2009)
Giuli, E., Tholen, W.: Openness with respect to a closure operator. Appl. Cat. Struct. 8, 487–502 (2000)
Holgate, D., Iragi, M., Razafindrakoto, A.: Topogenous and nearness structures on categories. Appl. Cat. Struct. accepted to appear
Holgate, D., Šlapal, J.: Categorical neighbourhood operators. Top. Appl. 158, 2356–2365 (2011)
Johnstone, P.: Stone spaces, vol. 370. Cambridge Univ. Press, New York (1983)
Lunna-Torres, J., Ochoa, C.O.C.: Interior operators and topological categories. Adv. Appl. Math. Sci., 189–206 (2011)
Picado, J., Pultr, A., Tozzi, A.: Locales, Categorical foundations. In: Pedicchio, M.C., Tholen, W. (eds.) Encyc. Math. Appl., 97, Cambridge Univ. Press, Cambridge, pp 103–163 (2004)
Razafindrakoto, A.: On coarse and fine neighbourhood operators. Top. Appl. 159, 3067–3079 (2012)
Razafindrakoto, A., Holgate, D.: Interior and neighbourhood. Top. Appl. 168, 144–152 (2014)
Šlapal, J.: Compactness and Convergence with respect to a neighbourhood operator. Collect. Math. 63, 123–137 (2012)
Tholen, W.: A categorical guide to separation, compactness and perfectness. Hom. Hom. Appl. 1, 147–161 (1999)
Tholen, W.: Closure operators and their middle-interchange law. Top. Appl. 158, 2437–2441 (2011)
Vorster, S.J.R.: Interior Operators in General categories. Quaest. Math. 23, 404–416 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Razafindrakoto, A., Holgate, D. A Lax Approach to Neighbourhood Operators. Appl Categor Struct 25, 431–445 (2017). https://doi.org/10.1007/s10485-016-9441-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-016-9441-3