On Coregular Closure Op erators Goncalo Gutierres Departamento de Matematica da Universidade de Coimbra Coimbra Portugal ggutcmatucpt Abstract Among closure op erators in the sense of Dikranjan and Giuli the regular ones have a relevant role and have b een widely investigated On the contrarythe coregular closure op erators were intro duced only recently in and they need to be further investigated In this pap er we study coregular closure op erators in connection connectednesses and disconnectednesses in the realm of top ological spaces and mo dules AMS sub ject classication B B E D Keywords closure op erator coregular closure op erator connectedness preradical torsion theory Intro duction Regular closure op erators were intro duced by Salbany in in the category of top ological spaces and have been investigated and used by several authors namely b ecause they play an imp ortant role in the study of epimorphisms They havealsobeen used in the context of the Diagonal Theorem that is the characterization of delta sub categories see for instance and The recent study of nabla sub categories by Clementino and Tholen led these authors to the denition of coregular closure op erator whichturnedouttoplay exactly the role of regular one in this context Besides some interesting examples presented in not muchisknown ab out these closure op erators even in the category of top ological spaces e In this pap er we investigate the b ehaviour nabla sub categories and their resp ectiv coregular closure op erators in the category T op of top ologoical spaces section and Mod of mo dules over aringR section R In T op we study in particular the least coregular closure op erators and obtain a prop er class of coregular closure op erators that do not form a chain Prop osition In Mod we show in Theorem that the regular and coregular closure op erators R are exactly the maximal and the minimal closure op erators dened by radicals and idemp otent radicals resp ectively Acknowledgment I thank Professor Maria Manuel Clementino for valuable dis cussion on the sub ject of this pap er Preliminaries We will rst intro duce some notions and techniques that will b e used througout The author acknowledges partial nancial assistance by the Centro de Matematica da Universidade de Coimbra and by Pro jects Praxis PCEXPMATACL and XXI MAT Factorization systems In the category T op of top ological spaces and continuous maps the class M of em b eddings has some sp ecial features that can be formulated in a categorical way For each space X the class MX of embeddings with co domain X can be preordered by where m M X n N X if there exists t M N such that n if m n the equivalence relation dened by m n t m Considering in MX and n m it is obtains that each equivalence class corresp onds exactly to an inclusion of a subspace of X In MX one can form arbitrary meets and so also arbitrary joins and the class M is stable under pullback Furthermore every morphism f X Y can be factorized as follows f X Y e a f X where a is an emb edding and e is a continuous surjection Moreover this factorization is unique up to isomorphism that is if f a e a e with a a emb eddings and e e surjections there exists an isomorphism h suchthath e e and a h a The direct m f image of M X under f is obtained by factorizing M X Y as ab ove while the n inverse image of N Y under f is exactly the pullback bred pro duct of N Y along f This construction can b e generalized straightforward to a general category X given two classes of morphisms E and M closed under comp osition and containing the iso morphisms of X one says that E M is a factorization system for morphisms in X if every X morphism has a unique up to isomorphism E Mfactorization That is for each morphism f X Y there exists e X M and m M Y in M such that f m e A factorization system is called proper if E is a class of epimorphisms and M is a class of monomorphisms Consequently E contains the regular epimorphisms and M the regular monomorphisms cf The category X is said to b e Mcomplete if X has Mpullbacks ie the pullbackof a morphism in M along any morphism exists and belongs to M and Mintersections ie X has limits of families of morphisms in M with common co domain which be long to M We remark that the Mcompleteness of X guarantees the existence of a factorization system E M for morphisms see Having in mind the behaviour in T op of the factorization describ ed ab ove in an Mcomplete category X with the factorization system E M a subobject of X X is a morphism m M X in M Denoting bysubX the class of sub ob jects of X subX is a p ossibly large complete lattice with its preorder dened as in the top ological setting Every morphism f X Y in X induces an imagepreimage adjunction f a f sub Y sub X with f n the pullback of n sub Y along f and f m the Mpart of the factorization of f m One always has m f f m and f f n n For more details on factorization systems see Closure Op erators From now on we work in an Mcomplete category X with nite limits and a prop er factorization system E M A closure op erator c of the category X with resp ect to the factorization system E M is given by a family of maps c sub X sub X such that X X X m c m for all m sub X X if m m then c m c m for all m m sub X X X f c m c f m for all m sub X and f X Y in X X Y Condition can equivalently b e expressed as c f n f c n for all n in sub Y X Y X Y in X and f A sub ob ject m of X is cclosed if c m m and it is cdense if c m X X X A closure op erator c is idempotent if cmiscclosed for every m Mandisweakly hereditary if j is cdense with m cm j m m The preorders of the classes sub X induce in a natural way a partial order in the conglomerate CLX of all closure op erators in X wrtE M that has meets and joins formed p ointwise For additional information on closure op erators see Regular and coregular closure op erators versus connectedness Given a closure op erator c in X an ob ject X of X is called cseparated if its diago nal X X X is cclosed and it is called cconnected if is X X X X cdense This w ay one denes the sub categories c of cseparated ob jects and rc of cconnected ob jects all the sub categories of X we consider are full and closed under isomorphisms and we denote its conglomerate by SU BX The ob jects that b elong to c rc are those whose diagonal is an isomorphism which are exactly the preter minal ob jects The and r assignments giverise to the functors op CLX SU BX r CLX SU BX where the partially ordered conglomerates CLX and SU BX are considered as cat egories On the other hand each sub category of X denes two sp ecial closure op erators a regular and a coregular closure op erator we describ e below Denitions Let A be a sub category of X The regular and coregular closure operators induced by A are lo cally dened by A reg m fh j h X A A A and hm g A A X A coreg mm fh j h A X A A and h mg A A X for every m sub X and every X X We remark that every regular closure op erator is idemp otent and every coregular closure op erator is weakly hereditary Regular closure op erators were intro duced by Salbany in with a dierent but equivalent description and were widely used in the literature Coregular clo were intro duced by Clementino and Tholen in in order to describ e sure op erators rsub categories A B A B Let A B be sub categories of X If A B then coreg coreg and reg reg hence reg and coreg maybe interpreted as functors Prop osition op The functor reg SU BX CLX is right adjoint to The functor coreg SU BX CLX is right adjoint to r Corollary Let A be a subcategory of X and c a closure operator in X Then A c a Areg and c reg A b Ac c reg rc A and c coreg a Arcoreg A b Arc c coreg From this prop osition one has that there is a bijection between coregular closure op erators and deltanabla sub categories Coregular closure op erators in T op In this section we will present examples of coregular closure op erators and rsub categories in the category of top ological spaces It was proved in that r and sub categories in T op extend disconnectednesses and connectednesses as studied by Arhangelski and Wiegandt Prop osition Let A be a subcategory of T op Then A r AfX Top j A A g A X g is constant g coreg A rreg l A fX Top j A A f X A f is constant g The sub categories of the typ e l A and r A are called leftconstant and right constant resp ectively and in the particular case of top ological spaces they are also called connectednesses and disconnectednesses The following examples were studied in Example Let k b e the Kuratowski closure op erator The sub category rk isthe class of Hausdor spaces and k is the class of irreducible spaces A top ological space X is irreducible if for U V X open sets and U V U or V The class C on of connected spaces is not the nabla sub category of the usual closure op erator k but as we will see is a nabla sub category Example Let conn be the connected component closure operator dened by S x where comp x is the connected comp onent of x The conn M comp X xM X X nabla sub category of conn is the sub category of connected spaces We