# Learners' Languages

arXiv: https://arxiv.org/abs/2103.01189 · HTML: https://mlsanity.com/paper/2103.01189/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 9 references that are in the map.
- First posted on arXiv 1 Mar 2021.

## Builds on (most cited first, 3 of 3)

- [Backprop as Functor: A compositional perspective on supervised learning](https://mlsanity.com/paper/1711.10455/) (2017)
- [Compositional game theory](https://mlsanity.com/paper/1603.04641/) (2016)
- [Directed Containers as Categories](https://mlsanity.com/paper/1604.01187/) (2016)

## Built on by (most cited first, 6 of 6)

- [Self-Revising Discovery Systems for Science: A Categorical Framework for Agentic Artificial Intelligence](https://mlsanity.com/paper/2606.01444/) (2026)
- [Dynamic Operads, Dynamic Categories: From Deep Learning to Prediction Markets](https://mlsanity.com/paper/2205.03906/) (2022)
- [Compositional Active Inference II: Polynomial Dynamics. Approximate Inference Doctrines](https://mlsanity.com/paper/2208.12173/) (2022)
- [The Topos of Transformer Networks](https://mlsanity.com/paper/2403.18415/) (2024)
- [Algebraic Dynamical Systems in Machine Learning](https://mlsanity.com/paper/2311.03118/) (2023)
- [Interactions that reshape the interfaces of the interacting parties](https://mlsanity.com/paper/2602.17917/) (2026)

## Abstract (arXiv)

In "Backprop as functor", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)→Learn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism LearncongPara(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor Amapsto AyA. Using the fact that (Poly,otimes) is monoidal closed, we show that a map A→ B in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type [AyA,ByB]. Finally, we review the fact that the category p-Coalg of dynamical systems on any p ∈ Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.