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(Lack of) Model Structures on the Category of Graphs

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Abstract

In this article, we study model structures on the category of finite graphs with \(\times \)-homotopy equivalences as the weak equivalences. We show that there does not exist an analogue of Strøm-Hurewicz model structure on this category of graphs. More interestingly, we show that this category of graphs with \(\times \)-homotopy equivalences does not have a model structure whenever the class of cofibrations is a subclass of graph inclusions.

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Notes

  1. A morphism that is both a weak equivalence and a cofibration is called an acyclic cofibration.

  2. A morphism that is both a weak equivalence and a fibration is called an acyclic fibration.

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Acknowledgements

We thank the anonymous referee for several suggestions which has improved this article.

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Authors and Affiliations

  1. Chennai Mathematical Institute, Chennai, India

    Shuchita Goyal

  2. Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India

    Rekha Santhanam

Authors
  1. Shuchita Goyal
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  2. Rekha Santhanam
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Correspondence to Rekha Santhanam.

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Communicated by Richard Garner.

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Goyal, S., Santhanam, R. (Lack of) Model Structures on the Category of Graphs. Appl Categor Struct 29, 671–683 (2021). https://doi.org/10.1007/s10485-021-09630-4

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