A homotopy category for graphs


We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call ‘spider moves.’ We then create a category by modding out by the 2-cells of our 2-category and use the spider moves to show that for finite graphs, this category is a homotopy category in the sense that it satisfies the universal property for localizing homotopy equivalences. We then show that finite stiff graphs form a skeleton of this homotopy category.

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The authors are grateful to Dr. Demitri Plessas for his previous forays into categorical graph theory and for his feedback. We also want to thank Dr. Jeffery Johnson for helping us with some of the terminology in this paper. Lastly, we want to thank the referees who pointed us in the direction of the relevant literature.

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Correspondence to Tien Chih.

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Chih, T., Scull, L. A homotopy category for graphs. J Algebr Comb (2020). https://doi.org/10.1007/s10801-020-00960-5

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  • Graph homomorphism
  • Homotopy
  • 2-category
  • Homotopy category
  • Skeleton of a category

Mathematics Subject Classification

  • 05C60
  • 55U35
  • 18D05