Bartosz Milewski • 6/30/2026

Actegories

This article delves into the concept of actegories, which are central to optics such as lenses, prisms, and traversals in programming. It begins by defining monoidal categories, including tensor products, associators, and unitors, with Haskell code examples. The article then introduces actegories as categories supporting the action of a monoidal category, describing how objects can be 'scaled' by monoidal objects. It covers the functorial action and coherence conditions, providing a technical foundation for understanding these structures in category theory and functional programming.

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#optics #Haskell #Category Theory