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The present chapter constitutes a self-contained introduction to the category theory of fuzzy simplicial sets, metric spaces and their various merging schemes and relationships (expanding results presented in Barth et al. (Fuzzy simplicial sets and their application to geometric data analysis, 2024), while including an introduction to all necessary preliminaries. We hope that this can help the interested reader to understand the details of the abstract theory underpinning UMAP and IsUMap, even without prior knowledge of category theory. Furthermore, we believe that the theory of fuzzy simplicial sets and metric spaces is interesting in their own right.

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Merging Fuzzy Simplicial Sets and Metric Spaces: A Category Theoretical Approach

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  • 出版模式：

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  • 发表日期：

    2025年

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  • Lukas Silvester Barth,
  • Hannaneh Fahimi,
  • Parvaneh Joharinad,
  • Jürgen Jost,
  • Janis Keck

摘要

The present chapter constitutes a self-contained introduction to the category theory of fuzzy simplicial sets, metric spaces and their various merging schemes and relationships (expanding results presented in Barth et al. (Fuzzy simplicial sets and their application to geometric data analysis, 2024), while including an introduction to all necessary preliminaries. We hope that this can help the interested reader to understand the details of the abstract theory underpinning UMAP and IsUMap, even without prior knowledge of category theory. Furthermore, we believe that the theory of fuzzy simplicial sets and metric spaces is interesting in their own right.