The reason I bored you with the Yoneda equipment was to eventually double back to Tannakian reconstruction. In a nutshell, Tannakian reconstruction lets us reconstruct the hom-set from a set of nat…

Working with double categories can be aplty summarized in a meme: Talk to me about sets without mentioning sets. We don’t talk about hom-sets, we talk about horizontal units. Secretly, we are…

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. $late…

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. …

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 ex…

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 multipli…

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 $latex (a, b)$, where $latex b = f a$. Similarly, a gr…

Previously: Profunctor Equipment in Haskell. The major advantage of string diagrams is that they provide surpisingly natural language for complex diagram manipulations. The fact that two traditiona…

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 th…

Notes on a nonexistent rule

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 interaction…

Previously: Identity Types. Let me first explain why the naive categorical model of dependent types doesn’t work very well for identity types. The problem is that, in such a model, any arrow …

Previously: Models of (Dependent) Type Theory. There is a deep connection between mathematics and programming. Computer programs deal with such mathematical objects as numbers, vectors, monoids, fu…

Previously: (Weak) Factorization Systems. It’s been known since Lambek that typed lambda calculus can be modeled in a cartesian closed category, CCC. Cartesian means that you can form product…

Previously (Weak) Homotopy Equivalences. An average function between sets, $latex f \colon A \to B$ is neither surjective nor injective. We can however isolate the two “failure modes” i…

Previously: Fibrations and Cofibrations. In topology, we say that two shapes are the same if there is a homeomorphism– an invertible continuous map– between them. Continuity means that …

We are used to thinking of a mapping as either being invertible or not. It’s a yes or no question. A mapping between sets is invertible if it’s both injective and surjective. It means t…

Previously: Subobject Classifier. In category theory, objects are devoid of internal structure. We’ve seen however that in certain categories we can define relationships between objects that …

A Mastodon instance for maths people. We have LaTeX rendering in the web interface!

Proviously Sieves and Sheaves. We have seen how topology can be defined by working with sets of continuous functions over coverages. Categorically speaking, a coverage is a special case of a sieve,…

There are many excellent AI papers and tutorials that explain the attention pattern in Large Language Models. But this essentially simple pattern is often obscured by implementation details and opt…

The yearly Advent of Code is always a source of interesting coding challenges. You can often solve them the easy way, or spend days trying to solve them “the right way.” I personally pr…

Previously: Covering Sieves. We’ve seen an intuitive description of presheaves as virtual objects. We can use the same trick to visualize natural transformations. A natural transformation can…