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Adjunction in the Absence of Identity

Abstract

We develop a bicategorical setup in which one can speak about adjoint 1-morphisms even in the absence of genuine identity 1-morphisms. We also investigate which part of 2-representation theory of 2-categories extends to this new setup.

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Data Availability Statement

Not applicable.

Notes

  1. 1.

    This also has to do with the terminology. There has been a suggestion that ‘weakly fiax’ should be ‘fax’ and ‘weakly fiat’ should be ‘fat’, which was dismissed by a majority of the authors of the current paper.

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Acknowledgements

This research was partially supported by the Swedish Research Council and Göran Gustafsson Stiftelse. We especially thank Marco Mackaay in conversation with whom the main idea behind this paper crystalized. We thank Gustavo Jasso for information about adjunction of semifunctors. We are grateful to the referee for very helpful comments.

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Correspondence to Xiaoting Zhang.

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Communicated by Nicola Gambino.

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Ko, H., Mazorchuk, V. & Zhang, X. Adjunction in the Absence of Identity. Appl Categor Struct (2021). https://doi.org/10.1007/s10485-021-09652-y

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Keywords

  • (Op)lax units
  • Adjunctions
  • Fiax categories
  • Bilax 2-representations

Mathematics Subject Classification

  • 18N10