Projective Quantales: A General View

Following B. Banaschewski’s work on so-called K-flat projective frames, we treat their non-commutative generalization, i.e. K-flat projective quantales. Besides a categorical definition of a K-flat projective quantale we also give an elementary characterization in terms of a binary relation on the quantale, and we prove that K-flat projective quantales are precisely the coalgebras for a certain comonad on the category of quantales. Finally we define and study a natural notion of K-coherentness for quantales. Financial Support of the Ministry of Education of the Czech Republic under the project MSM0021622409 is gratefully acknowledged.