# Actegories

DevFeed: [Actegories](<https://devfeed.tech/articles/actegories-28861.md>)

Original publisher: [Read original article](<https://bartoszmilewski.com/2026/06/30/actegories/>)

Author: Bartosz Milewski

Published: 2026-06-30T11:45:46Z

Content type: tutorial

Language: en

Sources: [Bartosz Milewski's Programming Cafe](<https://devfeed.tech/sources/bartosz-milewski-s-programming-cafe.md>)

Topics: [Programming](<https://devfeed.tech/topics/programming.md>), [Haskell](<https://devfeed.tech/topics/haskell.md>), [Code](<https://devfeed.tech/topics/code.md>)

Tags: [actegories](<https://devfeed.tech/tags/actegories.md>), [category-theory](<https://devfeed.tech/tags/category-theory.md>), [class](<https://devfeed.tech/tags/class.md>), [haskell](<https://devfeed.tech/tags/haskell.md>), [lens](<https://devfeed.tech/tags/lens.md>), [optics](<https://devfeed.tech/tags/optics.md>), [programming](<https://devfeed.tech/tags/programming.md>), [types](<https://devfeed.tech/tags/types.md>)

## AI overview

This tutorial introduces actegories, explaining their relationship to monoidal categories and their role in programming optics such as lenses, prisms, and traversals. It develops the concepts through Haskell models, including tensor products, unit objects, constraints, associators, unitors, and categorical actions.

## Source excerpt

Previously: Kan Extensions in Double Categories. In programming, actegories play a central role in optics: lenses, prisms, traversals, etc. To understand actegories, let's start with the definition of a monoidal category. Monoidal Category A monoidal category is a category equipped with a tensor product. A tensor product is a functor . We assume that this [...]