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PreprintCC BY 4.0arXiv

The category of nominal sets is locally monoidal closed

  • Fahimeh Bayeh
  • Peng Fu
  • Peter Selinger
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Abstract

The category Nom of nominal sets was proposed by Pitts and Gabbay as a setting for the semantics of abstract syntax with variable bindings. It is well-known that Nom is a topos, also known as the Schanuel topos, and in particular it follows that Nom is locally closed, i.e., every slice category Nom/X is cartesian-closed. In this paper, we show that Nom has a much stronger property: every monoidal (closed) structure on Nom induces a monoidal (closed) structure on all slice categories Nom/X. One monoidal closed structure of particular interest on Nom is the separated product A * B, whose right adjoint A -* B is called the separated function space. In particular, it follows that Nom is locally separatedly closed.

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Cite

BibTeXRIS
BibTeX
@misc{bayeh2026category,
  title = {{The category of nominal sets is locally monoidal closed}},
  author = {Fahimeh Bayeh and Peng Fu and Peter Selinger},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22551},
  eprint = {2609.22551},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22551},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The category of nominal sets is locally monoidal closed
AU  - Fahimeh Bayeh
AU  - Peng Fu
AU  - Peter Selinger
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.22551
UR  - https://arxiv.org/abs/2609.22551
AB  - The category Nom of nominal sets was proposed by Pitts and Gabbay as a setting for the semantics of abstract syntax with variable bindings. It is well-known that Nom is a topos, also known as the Schanuel topos, and in particular it follows that Nom is locally closed, i.e., every slice category Nom/X is cartesian-closed. In this paper, we show that Nom has a much stronger property: every monoidal (closed) structure on Nom induces a monoidal (closed) structure on all slice categories Nom/X. One monoidal closed structure of particular interest on Nom is the separated product A * B, whose right adjoint A -* B is called the separated function space. In particular, it follows that Nom is locally separatedly closed.
ER  -