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.

翻译：本文进一步推广了光学或“带孔电路”理论，使其涵盖前幺半范畴：即不满足交换律的幺半范畴。每个前幺半范畴通过中心态射的幺半子范畴嵌入，可导出一个效应范畴（即广义Freyd范畴）。我们引入“前效应范畴”概念，并证明前幺半范畴的光学结构恰好呈现此类结构。前效应范畴是效应范畴的非可表版本，类似于从幺半范畴到前幺半范畴的推广。我们将Day的经典结论推广至此框架，证明范畴上的前效应结构与自由锥余完备化上的效应结构存在等价关系。此外，我们证明了前效应范畴等价于前强前幺半群。