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Homotopy and Hom Construction in the Category of Finite Hypergraphs

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Abstract

We define notions of a weak homotopy for finite hypergraphs and an exponential hypergraph with a right adjoint to the categorical product of finite hypergraphs, and investigate a connection between the weak homotopy and the exponential hypergraph. Then we discuss a Hom construction associated to a pair of finite hypergraphs and prove that the homotopy of hypergraph homomorphisms, defined in (Grigor’Yan et al. in Topol Appl 267:106877, 2019), could be characterized by properties of the Hom construction. In addition, we establish some properties of Hom constructions involving the categorical product of finite hypergraphs. As an application we show that the homotopy of d-colorings of a simple hypergraph \({\mathcal {H}}\) could be characterized by properties of Hom constructions associated to the maximum simple hypergraph and \({\mathcal {H}}\).

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Funding

This research was supported by the National Natural Science Foundations of China (No. 11771116) and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2022L266).

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Authors and Affiliations

  1. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

    Conglei Zhang & Yanying Wang

  2. College of Arts and Sciences, Shanxi Agricultural University, Jinzhong, 030801, People’s Republic of China

    Conglei Zhang

  3. School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, 030603, People’s Republic of China

    Zhiguo Zhang

  4. College of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, 050061, People’s Republic of China

    Wei Dai

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  1. Conglei Zhang
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All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to Yanying Wang.

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Zhang, C., Wang, Y., Zhang, Z. et al. Homotopy and Hom Construction in the Category of Finite Hypergraphs. Graphs and Combinatorics 39, 78 (2023). https://doi.org/10.1007/s00373-023-02672-6

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  • DOI: https://doi.org/10.1007/s00373-023-02672-6

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