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Isbell Adjunctions and Kan Adjunctions via Quantale-Enriched Two-Variable Adjunctions

Abstract

It is shown that every two-variable adjunction in categories enriched in a commutative quantale serves as a base for constructing Isbell adjunctions between functor categories, and Kan adjunctions are precisely Isbell adjunctions constructed from suitable associated two-variable adjunctions. Representation theorems are established for fixed points of these adjunctions.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. 1.

    Strictly speaking, “fixed point” should read “pseudo-fixed point” here, since \(a\in \mathsf{Fix}(h)\) satisfies \(ha\cong a\) instead of \(ha=a\).

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Acknowledgements

The authors would like to thank Professor Hongliang Lai and Professor Dexue Zhang for helpful discussions. The authors would also like to thank the anonymous referee for valuable comments and suggestions which significantly improve the presentation of the paper.

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Correspondence to Lili Shen.

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This work was supported by National Natural Science Foundation of China (No. 12071319).

Communicated by Maria Manuel Clementino.

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Shen, L., Tang, X. Isbell Adjunctions and Kan Adjunctions via Quantale-Enriched Two-Variable Adjunctions. Appl Categor Struct (2021). https://doi.org/10.1007/s10485-021-09654-w

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Keywords

  • Quantale
  • Quantale-enriched category
  • Two-variable adjunction
  • Isbell adjunction
  • Kan adjunction

Mathematics Subject Classification

  • 18D20
  • 18A40
  • 18F75