By Dikranjan D.N., Tholen W.

This publication presents a finished express concept of closure operators, with functions to topological and uniform areas, teams, R-modules, fields and topological teams, as good as partly ordered units and graphs. particularly, closure operators are used to offer suggestions to the epimorphism and co-well-poweredness challenge in lots of concrete different types. the fabric is illustrated with many examples and routines, and open difficulties are formulated which may still stimulate extra learn. viewers: This quantity may be of curiosity to graduate scholars researchers in lots of branches of arithmetic and theoretical desktop technological know-how. wisdom of algebra, topology, and the easy notions of type concept is believed.

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7, except full additivity, directed additivity, and minimality. C (Characterization of idempotency and weak he reditariness) Prove for a closure operator C of X with respect to M: (a) (b) is idempotent if and only if for all in E MIX , cX(m) = n{n E M/X : n > m and n is C-closed). C is weakly hereditary if and only if for all in E MIX , C XEX XEX cX(m)25 (Note that it is not necessary here to assume X to be M-complete). D (a) (Defining closure operators from given closed su6objects) In an M-complete category X, let C C M be a subclass containing all isomorphisms, closed under composition with isomorphisms, and stable under pullback.

Then X is M-complete if and only if M is stable under pullback and multiple pullback. COROLLARY One has to show that any class-indexed family (mi)iEJ in MIX has an Proof intersection if X is complete and M-wellpowered. J . Therefore, the multiple pullback of (mj)JEj (which exists since X is assumed to have all small limits) serves also as a multiple pullback of (mi)WET 0 The rest of the proof is trivial. EXAMPLES The right M-factorization of a sink (1) class of injective maps is given by (fi : Xi -+Y)iE' in Set with M the Xi-M=Ufi(Xi)-Y.

Then X is finitely M-complete. COROLLARY Proof By the Proposition and the Theorem, M is stable under pullback. Since 0 X has pullbacks, M-pullbacks exist in X. 8 M-subobjects of M-subobjects For M-subobjects m : M -+ N and n : N -+ X , the composite n m : M , X should be an M-subobject. 7, for a good reason: finite M-completeness does not imply that M is closed under composition. EXAMPLE In the category Grp of groups, let M be the class of those injective homomorphisms f : G -+ H for which f (G) is normal in H .