Abstract
We define the concept of fuzzy subobject of a "good" object in an arbitrary concrete category. The approach, called geometric fuzzification, is based upon an appropriate concept of fuzzy point. These fuzzy points are defined on metric spaces which allow the introduction of a geometric notion of convexity. Busemann G-spaces (in particular, complete Riemannian manifolds) are recognized as such metric spaces. As an example, we produce a definition of a fuzzy geometric k-flat (i. e. k-dimensional linear submanifold) in the Euclidean space Rn. Fuzzy parametrized k-flats are discussed briefly.
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