Tabulation Tribulations
The article discusses the concept of a tabulation in a double category, which is a way of representing a profunctor as a 0-cell equipped with two projections. A tabulation is defined using its universal property, which involves a 1-dimensional and a 2-dimensional condition. The article explains how these conditions work and how they can be used to construct a mapping from a horizontal 1-cell to a tabulation.
- ▪A profunctor can be viewed as a proof-relevant relation.
- ▪A tabulation of a horizontal arrow is a 0-cell equipped with two projections.
- ▪The universal property of a tabulation involves a 1-dimensional and a 2-dimensional condition.
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Record
| Original publisher | Bartosz Milewski's Programming Cafe |
| Canonical URL | https://bartoszmilewski.com/2026/05/23/tabulation-tribulations/ |
| Publication time | Fri, 26 Jun 2026 07:44:09 +0000 |
| Retrieval time | 2026-06-26T08:07:31.019Z |
| Last seen | 2026-06-26T08:07:31.019Z |
| Headline source | Publisher (no WeSearch rewrite) |
| Excerpt source | publisher body |
| Excerpt method | First ~120 words (~800 chars) of extracted publisher body, fair-use limited. |
| Summary | WeSearch · cerebras-chat (WeSearch summarizer) |
| Summary source text | contentText |
| Citation coverage | Summary is a WeSearch-generated derivative; primary citation is the original publisher URL. |
| Cluster | None |
| Cluster logic | Not yet clustered, or no peer story found in the clustering window. |
| Ranking reason | Story pages are not engagement-ranked. Hub feeds use recency, with optional source-diversified chronological ordering (cap consecutive stories per source). No personalized ranking. |
| Publisher visit | Yes — open original |
| Substitutes article? | No — link-out required for full text |
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| Indexing | May the item be indexed (stored, ranked, made findable)? | Allowed |
| Snippet | May a short excerpt of the publisher's text be shown? | Allowed |
| AI summary | May WeSearch generate its own short summary of the article? | Limited |
| Retrieval / RAG | May the content be exposed for third-party retrieval-augmented generation? | Not asserted |
| Model training | May the content be used to train AI models? | Not asserted |
| Commercial reuse | May the content be reused commercially? | Not permitted |
Basis: Derived from the published RSS/Atom feed. Contact: [email protected]. Reviewed: 2026-07-24.
Opening excerpt (first ~120 words) tap to expand
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 of a profunctor is a triple, which contains two objects and a proof, or a witness, that they are related. The witness in this case is any element of the set . If the set is empty, it means that the objects are unrelated. Such triples form a category, which is often called the category of elements of a profunctor. Given a profunctor , an object in the category of elements is a triple . We interpret as a witness that is related to .
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Excerpt limited to ~120 words for fair-use compliance. The full article is at Bartosz Milewski's Programming Cafe.
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