The motivation of the work stems from the dimensionality reduction algorithm UMAP (``Uniform Manifold Approximation and Projection'' Algorithm) which was introduced in 2018 by L. McInnes, J. Healy and J. Melville. We will address its interpretation as a special case of manifold approximation using fuzzy simplicial sets, which sets it appart from the other manifold learning methods. The definition of a fuzzy simplicial set will arise by gradual generalization of simplicial complexes, using the language of category theory. By generalization of the singular set and geometric realization functors to the categoriess ${\cal F}in$-$s{\cal F}uzz$ of bounded fuzzy simplicial sets and ${\cal F}in{\cal EPM}et$ of finite extended pseudo-metric spaces we will describe the manifold approximation in a functorial way and discuss its implementation in the UMAP algorithm.
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