Abstract
A new definition of connectedness of an object in a category with respect to a closure operator is given. It is shown that many of the classical results about connectedness of topological spaces, under mild conditions, hold in an arbitrary category. In particular it is shown that the image of a connected object is connected; that the union and the product of connected objects are connected. Several illustrative examples are provided.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adamek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. Wiley, New York (1990)
Castellini, G.: Connectedness with respect to a closure operator. Appl. Categ. Struct. 9, 285–302 (2001)
Castellini, G.: Categorical Closure Operators. Birkhäuser, Boston (2003)
Castellini, G., HajekClosure, D.: Operators and connectedness. Topol. Appl. 55, 29–45 (1994)
Castellini, G., Holgate, D.: A link between two connectedness notions. Appl. Categ. Struct. 11, 473–486 (2003)
Clementino, M.M.: On connectedness via closure operator. Appl. Categ. Struct 9, 539–556 (2001)
Clementino, M.M., Tholen, W.: Separation versus connectedness. Topol. Appl. 75, 143–181 (1997)
Dikranjan, D., Tholen, W.: Categorical Structure of Closure Operators. Kluwer Academic Publishers, Amsterdam (1995)
Enochs, E.E., Jenda, O.M.G.: Relative Homological Algebra. Walter de Gruyter, Berlin (2000)
Maclane, S., Moerdijk, I.: Sheaves in Geometry and Logic, A First Introduction to Topos Theory. Springer, Berlin (1992)
Mousavi, S.Sh, Hosseini, S.N.: Quasi right factorization structures as presheaves. Appl. Categ. Struct. 19, 741–756 (2011). https://doi.org/10.1007/s10485-010-9242-z
Preuss, G.: Relative connectedness and disconnectedness in topological categories. Quaest. Math. 2, 297–306 (1977)
Slapal, J.: Another approach to connectedness with respect to a closure operator. Appl. Categ. Struct. 17, 603–612 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rahim Zaare-Nahandi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shir Ali Nasab, A.R., Hosseini, S.N. Connectedness in a Category. Bull. Iran. Math. Soc. 46, 1195–1210 (2020). https://doi.org/10.1007/s41980-019-00320-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-019-00320-5