Optics for Premonoidal Categories

Last updated Dec 7, 2024

Abstract. We further the theory of optics or “circuits-with-holes” to encompass premonoidal categories: monoidal categories without the interchange law. Every premonoidal category gives rise to an effectful category (i.e. a generalised Freyd-category) given by the embedding of the monoidal subcategory of central morphisms. We introduce “pro-effectful” categories and show that optics for premonoidal categories exhibit this structure. Pro-effectful categories are the non-representable versions of effectful categories, akin to the generalisation of monoidal to promonoidal categories. We extend a classical result of Day to this setting, showing an equivalence between pro-effectful structures on a category and effectful structures on its free conical cocompletion. We also demonstrate that pro-effectful categories are equivalent to prostrong promonads.

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@inproceedings{roman23:opticsForPremonoidal,
  author       = {James Hefford and
                  Mario Rom{\'{a}}n},
  editor       = {Sam Staton and
                  Christina Vasilakopoulou},
  title        = {Optics for Premonoidal Categories},
  booktitle    = {Proceedings of the Sixth International Conference on Applied Category
                  Theory 2023, {ACT} 2023, University of Maryland, 31 July - 4 August
                  2023},
  series       = {{EPTCS}},
  volume       = {397},
  pages        = {152--171},
  year         = {2023},
  url          = {https://doi.org/10.4204/EPTCS.397.10},
  doi          = {10.4204/EPTCS.397.10},
  timestamp    = {Sun, 04 Aug 2024 19:45:39 +0200},
  biburl       = {https://dblp.org/rec/journals/corr/abs-2305-02906.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}