Idempotency of Extensions via the Bicompletion

Autor: Hans-Peter A. Künzi, Guillaume C. L. Brümmer
Rok vydání: 2006
Předmět:
Zdroj: Applied Categorical Structures. 15:499-509
ISSN: 1572-9095
0927-2852
DOI: 10.1007/s10485-006-9028-5
Popis: Let Top 0 be the category of topological T 0-spaces, QU 0 the category of quasi-uniform T 0-spaces, T : QU 0 → Top 0 the usual forgetful functor and K : QU 0 → QU 0 the bicompletion reflector with unit k : 1 → K. Any T-section F : Top 0 → QU 0 is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU 0 is not stable under K.
Databáze: OpenAIRE
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