Bartosz Milewski's Programming Cafe

You may think of Tannakian Reconstruction as an example of redundant encoding. It lets you replace a simple hom-set with a much more complex end that is taken over an entire functor category. Why would anyone want to do it? The answer is simple: compositi

Two friends, Alice and Bob, live in the same city, but on the opposite sides of a wide river. Every night, Bob looks at the lights on the other side and tries to guess, which one belongs to Alice. They come up with a clever arrangement: Alice will turn on

I was originally attracted to category theory when trying to understand Haskell optics. I was puzzled by the van Laarhoven's functor representations and Kmett's use of Tambara modules. By playing Tetris with the Yoneda lemma I was able to make some progre

Previously: Kan extensions in Haskell. In a double category that is also a proarrow equipment, we have the ability to bend arrows. In particular, in the definition of the counit of the right Kan extension: we can bend the vertical arrow, replacing it with

Previously: Tabulation Tribulations. If you think of functor composition as a form of multiplication, Kan extensions are an attempt to construct inverses of this multiplication. But unlike multiplication, composition is not symmetric, so we have extension

Previously: Bending, Yanking, and Cartesian Squares in Double Categories. We all know what a graph of a function is: it's a set of pairs , where . Similarly, a graph of a relation is a set of pairs where is related to . A profunctor can be viewed as a pro

Previously: Profunctor Equipment in Haskell. The major advantage of string diagrams is that they provide surprisingly natural language for complex diagram manipulations. The fact that two traditional diagrams are equal can be often described as a permissi

Previously: Profunctor Equipment. To make things more palatable for programmers, I decided to provide a toy implementation of some of the equipments in Haskell. The advantage of this encoding is that it can be verified by the compiler, and I still trust t

The fundamental premise of category theory is that it's possible to fully capture the nature of objects by describing their interactions with other objects of the same type. Those interactions are encoded using morphisms: arrows between objects. What abou

Previously: Modeling Identity Types. On first viewing, the identity type seems odd. Does it make sense to replace the traditional yes/no equality predicate with an elaborate type of equality proofs? In fact the father or modern type theory Martin Löf had