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PreprintCC BY 4.0arXiv

Twisted Bicategorical Shadows and Traces

  • Zhonghui Sun
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Abstract

Bicategorical shadows provide a categorical framework that encompasses Hochschild homology and topological Hochschild homology (THH), encodes their Morita invariance, and extends the notion of trace from symmetric monoidal categories to bicategories. Certain equivariant variants, such as Cn-twisted THH, do not fit into the ordinary shadow framework. We introduce bicategories with G-twisting data and show that every ordinary shadow on such a bicategory induces, for each g∈ G, a g-twisted shadow on the associated bicategory of G-twists. These g-twisted shadows are invariant under G-Morita equivalence; examples include Cn-twisted THH and twisted Hochschild homology of Cn-Green functors. We further define a g-twisted bicategorical trace that recovers the g-twisted Hattori--Stallings trace. For Cn-Green functors, the orbitwise g-twisted Hattori--Stallings traces assemble into a morphism of Cn-Mackey functors that, at the orbit Cn/Cn, recovers the degree-zero twisted Dennis trace.

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Cite

BibTeXRIS
BibTeX
@misc{sun2026twisted,
  title = {{Twisted Bicategorical Shadows and Traces}},
  author = {Zhonghui Sun},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04660},
  eprint = {2609.04660},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04660},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Twisted Bicategorical Shadows and Traces
AU  - Zhonghui Sun
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04660
UR  - https://arxiv.org/abs/2609.04660
AB  - Bicategorical shadows provide a categorical framework that encompasses Hochschild homology and topological Hochschild homology (THH), encodes their Morita invariance, and extends the notion of trace from symmetric monoidal categories to bicategories. Certain equivariant variants, such as C_n-twisted THH, do not fit into the ordinary shadow framework. We introduce bicategories with G-twisting data and show that every ordinary shadow on such a bicategory induces, for each g∈ G, a g-twisted shadow on the associated bicategory of G-twists. These g-twisted shadows are invariant under G-Morita equivalence; examples include C_n-twisted THH and twisted Hochschild homology of C_n-Green functors. We further define a g-twisted bicategorical trace that recovers the g-twisted Hattori--Stallings trace. For C_n-Green functors, the orbitwise g-twisted Hattori--Stallings traces assemble into a morphism of C_n-Mackey functors that, at the orbit C_n/C_n, recovers the degree-zero twisted Dennis trace.
ER  -