On multivalued topologies on $L$-powersets of multivalued sets

Given an $M$-valued equality $E\colon X\times X\to M$ on a set $X$, we extend it to the $M$-valued equality $\mathcal E\colon L^X\times L^X\to M$ on the $L$-powerset $L^X$ of $X$, where $L$ is a complete sublattice of a GL-monoid $M$. As a result, we come to a category $\mathbf{SET}(M, L)$ whose objects are quadruples $(X,E,L^X,\mathcal E)$. This category serves as a ground category for the category $L\text{-}\mathbf{TOP}(M)$ of $(L,M)$-valued topological spaces and some of its subcategories, which are the main subject of this paper. In particular, as special cases, we obtain here Chang–Goguen, Lowen, Kubiak–Šostak, and some other known categories related to fuzzy topology.