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Regular closure operators

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Abstract

In an 〈E,M〉-categoryX for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms inM to factor through the “lattice” of all closure operators onM, and to factor through certain sublattices. This leads to the notion ofregular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplün-Röhrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. (One of these factorizations constituted the main result of an earlier paper by the same authors). Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class ofX-objects as separated objects. The same relation that is used in Salbany's relative dominion construction induces classical regular closure operators as described above. Many other types of closure operators can be obtained by this technique; particular instances of this are the idempotent and modal closure operators that in a Grothendieck topos correspond to the Grothendieck topologies.

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Dedicated to Professor Dieter Pumplün, on his 60th birthday

Research partially supported by the Faculty of Arts and Sciences, University of Puerto Rico, Mayagüez Campus during a sabbatical visit at Kansas State University.

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Castellini, G., Koslowski, J. & Strecker, G.E. Regular closure operators. Appl Categor Struct 2, 219–244 (1994). https://doi.org/10.1007/BF00878099

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Mathematics Subject Classifications (1991)

  • Primary: 18A32
  • Secondary: 06A15, 54B30

Key words

  • Galois connection
  • polarity
  • closure operator
  • separated object
  • dense morphism
  • composable pair of morphisms
  • factorization structure for sinks