Categories isomorphic to L-FTOP

In this paper, our purpose is twofold. First, we deal with the local structure of L-fuzzy topology. Hence, modifying the idea of quasi-coincident neighborhood in L-topology, we concretely create an axiomatic system of neighborhood in L-fuzzy topology. Thus we find the category L-FQN of topological L-fuzzy quasi-coincident neighborhood spaces and appropriate continuous maps, which is isomorphic to the category L-FTOP, where L is fuzzy lattice. Further we prove that the category L-FQN is concretely bireflective in another category L-PrFQN constructed in the paper and L-PrFQN is topological category over SET. Secondly, to establish the relation between L-FTOP and L-TOP, we construct two categories L-AITOP and LaTQN. It is proved that L-AITOP is isomorphic to L-FTOP, in particular, when L is fuzzy lattice with locally multiplicative property, LaTQN is isomorphic to L-FTOP.