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arXiv · 2305.02906

Optics for Premonoidal Categories

James Hefford·Mario Román
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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 tight cocompletion. We also demonstrate that pro-effectful categories are equivalent to prostrong promonads.

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James Hefford, Mario Román. 2023-12-14. Optics for Premonoidal Categories. https://doi.org/10.4204/eptcs.397.10

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