  Closure operators are very useful tools in several areas of classical
mathematics and in general category theory. In fuzzy set theory, fuzzy closure
operators have been studied by G. Gerla (1966). These works generally define a
fuzzy subset as a mapping from a set X to the real unit interval, as a complete
and complemented lattice. More recently, Y. C. Kim (2003), F. G. Shi (2009), J.
Fang and Y. Yue (2010) propose theories of fuzzy closure systems and fuzzy
closure operators in a more general settings, but still using complemented
lattices. The aim of this paper is to propose a more general theory of fixed
and variable-basis fuzzy closure operators, employing both categorical tools
and the lattice theoretical fundations investigated by S. E. Rodabaugh (1999),
where the lattices are usually non-complemented. Besides, we construct
topological categories in both cases.
