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PreprintarXiv licencearXiv

Transfer of abelian model structures to equivariant categories and homotopy squares

  • Zhenxing Di
  • Liping Li
  • Li Liang
  • Guoliang Tang
  • Rongmin Zhu
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Abstract

Let G be a finite group acting on a Grothendieck category π’œ with enough projectives, such that |G| is invertible in π’œ. We prove a general lifting theorem for abelian model structures from π’œ to its equivariant category π’œG, and establish a triangle equivalence up to retracts between the corresponding homotopy categories. We also construct a commutative square whose horizontal functors are triangle equivalences and whose vertical comparison functors are triangle equivalences up to retracts. This square relates derived functors on the lifted equivariant model categories to the equivariantizations of the derived functors on the original homotopy categories. In the module category setting, we illustrate the above results using the PGF Hovey triples, and apply them to homotopy squares induced by a Frobenius bimodule and by a stable equivalence of adjoint type.

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Cite

BibTeXRIS
BibTeX
@misc{di2026transfer,
  title = {{Transfer of abelian model structures to equivariant categories and homotopy squares}},
  author = {Zhenxing Di and Liping Li and Li Liang and Guoliang Tang and Rongmin Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.08141},
  eprint = {2608.08141},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.08141},
}
RIS
TY  - UNPB
TI  - Transfer of abelian model structures to equivariant categories and homotopy squares
AU  - Zhenxing Di
AU  - Liping Li
AU  - Li Liang
AU  - Guoliang Tang
AU  - Rongmin Zhu
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2608.08141
UR  - https://arxiv.org/abs/2608.08141
AB  - Let G be a finite group acting on a Grothendieck category π’œ with enough projectives, such that |G| is invertible in π’œ. We prove a general lifting theorem for abelian model structures from π’œ to its equivariant category π’œ^G, and establish a triangle equivalence up to retracts between the corresponding homotopy categories. We also construct a commutative square whose horizontal functors are triangle equivalences and whose vertical comparison functors are triangle equivalences up to retracts. This square relates derived functors on the lifted equivariant model categories to the equivariantizations of the derived functors on the original homotopy categories. In the module category setting, we illustrate the above results using the PGF Hovey triples, and apply them to homotopy squares induced by a Frobenius bimodule and by a stable equivalence of adjoint type.
ER  -