- Author
- Bartosz Milewski
- Published
- 30 June 2026
- Kind
- Personal blog post, category theory with Haskell code
- Original
- bartoszmilewski.com/2026/06/30/actegories/
- Subject
- Programming languages and types
What it is
Bartosz Milewski writes about category theory and Haskell. The post follows an earlier one on Kan extensions. It defines the concepts the post says optics rely on.
What it says
- Milewski says actegories play a central role in optics such as lenses, prisms and traversals.
- A monoidal category has a tensor product that is associative and unital up to isomorphism.
- The post models a monoidal category in Haskell as a class over a bifunctor.
- An actegory is a category that supports an action of a monoidal category.
- The post simplifies by making the action a bifunctor and letting it determine the tensor.
- A traversable functor gives an example of a monoidal functor between non-trivial actegories.
What evidence it gives
The post gives definitions and Haskell class declarations. It shows instances for pairs as the tensor and for a traversable functor. It links to the full Haskell code but offers no proofs or tests.
Interests and limits
The formulas were lost in text extraction, so only prose and code were read. The post is a short setup piece. It does not show how optics are built from actegories.