caa a22 4500 467882819 CHVBK 20180406152725.0 cr unu---uuuuu 170328e20060401xx s 000 0 eng 10.1007/s10485-006-9016-9 doi (NATIONALLICENCE)springer-10.1007/s10485-006-9016-9 A Relation Between Closure Operators on a Small Category and Its Category of Presheaves [Elektronische Daten] [S. Hosseini, S. Mousavi] In this paper we show that each factorization structure ${\mathcal {M}}$ on a small category ${\mathcal {X}}$ , satisfying certain conditions, yields a presheaf ${{\boldsymbol{M}}}$ on ${\mathcal {X}}$ and a morphism of presheaves $${\mathbf{m}}:\Omega \xrightarrow{.}{\mathbf{M}}$$ . We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on ${\mathcal {X}}$ (b) subobjects of ${\boldsymbol{M}}$ (c) morphisms from ${\boldsymbol{M}}$ to ${\boldsymbol{\Omega}}$ (d) weak Lawvere-Tierney topologies (e) weak Grothendieck topologies (f) closure operators on $Sets^{{\mathcal {X}}^{op}}$ . Springer Science+Business Media B.V., 2006 factorization structure nationallicence Grothendieck topology nationallicence (idempotent, modal) closure operator nationallicence Lawvere-Tierney topology nationallicence Hosseini S. Mathematics Department, Shahid Bahonar University of Kerman, Kerman, Iran aut Mousavi S. Mathematics Department, Vali-ASR University of Rafsandjan, Rafsandjan, Iran aut Applied Categorical Structures Kluwer Academic Publishers 14/2(2006-04-01), 99-110 0927-2852 14:2<99 2006 14 10485 https://doi.org/10.1007/s10485-006-9016-9 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10485-006-9016-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Hosseini S. Mathematics Department, Shahid Bahonar University of Kerman, Kerman, Iran aut NATIONALLICENCE 700 1- Mousavi S. Mathematics Department, Vali-ASR University of Rafsandjan, Rafsandjan, Iran aut NATIONALLICENCE 773 0- Applied Categorical Structures Kluwer Academic Publishers 14/2(2006-04-01), 99-110 0927-2852 14:2<99 2006 14 10485 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer