# The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits

arXiv: https://arxiv.org/abs/1305.0297 · HTML: https://mlsanity.com/paper/1305.0297/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 May 2013.

## Built on by (most cited first, 9 of 9)

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- [On The Universality of Diagrams for Causal Inference and The Causal Reproducing Property](https://mlsanity.com/paper/2207.02917/) (2022)
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- [Quantifying analogy of concepts via ologs and wiring diagrams](https://mlsanity.com/paper/2402.01020/) (2024)
- [A Probabilistic Generative Model of Free Categories](https://mlsanity.com/paper/2205.04545/) (2022)
- [A Categorical Framework for Modeling with Stock and Flow Diagrams](https://mlsanity.com/paper/2211.01290/) (2022)
- [A Formalization of Operads in Coq](https://mlsanity.com/paper/2303.08894/) (2023)
- [Interactions that reshape the interfaces of the interacting parties](https://mlsanity.com/paper/2602.17917/) (2026)
- [The Sensation Modulating Network:Haltability as the architectural ground for object-directed phenomenology](https://mlsanity.com/paper/2605.26856/) (2026)

## Abstract (arXiv)

Wiring diagrams, as seen in digital circuits, can be nested hierarchically and thus have an aspect of self-similarity. We show that wiring diagrams form the morphisms of an operad mcT, capturing this self-similarity. We discuss the algebra Rel of mathematical relations on mcT, and in so doing use wiring diagrams as a graphical language with which to structure queries on relational databases. We give the example of circuit diagrams as a special case. We move on to show how plug-and-play devices and also recursion can be formulated in the operadic framework as well. Throughout we include many examples and figures.