Homotopy theory in additive categories with suspensions

Abstract

We develop a homotopy theory in additive categories endowed with additive endofunctors, analogous to Quillen’s model categories theory. As applications, we show that Iyama–Yoshino triangulated subfactor categories can be modeled; we prove that Verdier quotients can be realized as triangulated subfactors in some cases; we construct the homotopy theory of Hovey triples in arbitrary exact categories.

Keywords:

• Exact model categories
• homotopy theory
• triangulated categories
• Verdier quotients

2020 Mathematics Subject Classification:

• 18E35
• 18E10
• 18E30

Acknowledgments

I would like to thank Henning Krause, Xiao-Wu Chen, Yu Ye, Guodong Zhou, Ming Lu for their helpful discussions and suggestions. I would especially like to thank Yan Lu for her translating [25] into English. The author would like to thank the referee for his/her valuable comments and the improvement of the main results. The warmhearted referee does not only point out the equivalence of Remark 9.4, but also gives a proof.

Notes

1 It is a contraction of pre, right and left.

2 The author would like to thank the referee for giving this equivalence and its proof.

3 The author would like to thank Ming Lu for the helpful discussions of this example.

Additional information

Funding

The author was supported by National Natural Science Foundation of China [Nos. 11671174 and 11571329].