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.
@inproceedings{premonoidalOptics23,
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},
doi = {10.4204/EPTCS.397.10},
timestamp = {Sun, 04 Aug 2024 19:45:39 +0200},
}