Tannakian Reconstruction in a Yoneda Equipment

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…

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Yoneda Lemma in Double Categories

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…

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Profunctor Optics

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…

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Tambara Equipment

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. …

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Kan Extensions in Double Categories

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…

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Kan Extensions in Haskell

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…

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Tabulation Tribulations

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…

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Bending, Yanking, and Cartesian Squares in Double Categories

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…

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Profunctor Equipment in Haskell

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…

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'Evacuate': Does it refer to people or places?

Notes on a nonexistent rule

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Profunctor Equipment

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…

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Modeling Identity Types

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 …

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Identity Types

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…

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Models of (Dependent) Type Theory

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…

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(Weak) Factorization Systems

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…

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(Weak) Homotopy Equivalences

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 …

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Fibrations and Cofibrations

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…

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Subfunctor Classifier

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 …

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Mathstodon

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

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Subobject Classifier

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,…

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Understanding Attention in LLMs

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…

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Data.Functor.Monoidal

hackage.haskell.org