Bartosz Milewski
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: composition! Morphisms on the left compose according...
Bartosz Milewski
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 progress, attacking more and more esoteric...
Bartosz Milewski
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 its horizontal conjoint . In a profunctor...
Bartosz Milewski
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 proof-relevant relation. So a graph […]
Bartosz Milewski
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 permission to bend, yank, or pinch strings in...
Bartosz Milewski
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 about categories themselves? We define a...