Serving as a generalization of many examples of fuzzy algebraical systems, we introduce fuzzy categories and show that categories formed by fuzzy categories are topological. By using this, we show results concerning the existence of limits and colimits in such categories. We apply these results to the categories of fuzzy sets, fuzzy categories, fuzzy groupoids, fuzzy monoids, fuzzy groups, fuzzy abelian groups and fuzzy ordered sets. Thereafter, we determine the complete ordered lattice structure of the collection of grade maps on some finite categories, in particular on cyclic groups of prime power order. We use this in the end of the article to construct grade maps on p-adic groups.

topological functor, fuzzy set, fuzzy group, fuzzy module.