# Lenses and Learners

arXiv: https://arxiv.org/abs/1903.03671 · HTML: https://mlsanity.com/paper/1903.03671/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Mar 2019.

## Built on by (most cited first, 11 of 11)

- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Pooling Image Datasets With Multiple Covariate Shift and Imbalance](https://mlsanity.com/paper/2403.02598/) (2024)
- [Deep Learning with Parametric Lenses](https://mlsanity.com/paper/2404.00408/) (2024)
- [Space-time tradeoffs of lenses and optics via higher category theory](https://mlsanity.com/paper/2209.09351/) (2022)
- [Towards a Categorical Foundation of Deep Learning: A Survey](https://mlsanity.com/paper/2410.05353/) (2024)
- [A Probabilistic Generative Model of Free Categories](https://mlsanity.com/paper/2205.04545/) (2022)
- [Algebraic Dynamical Systems in Machine Learning](https://mlsanity.com/paper/2311.03118/) (2023)
- [Higher Gauge Flow Models](https://mlsanity.com/paper/2507.16334/) (2025)
- [Disentangling Hyperedges through the Lens of Category Theory](https://mlsanity.com/paper/2510.16289/) (2025)
- [Modeling Topological Impact on Node Attribute Distributions in Attributed Graphs](https://mlsanity.com/paper/2602.01454/) (2026)

## Abstract (arXiv)

Lenses are a well-established structure for modelling bidirectional transformations, such as the interactions between a database and a view of it. Lenses may be symmetric or asymmetric, and may be composed, forming the morphisms of a monoidal category. More recently, the notion of a learner has been proposed: these provide a compositional way of modelling supervised learning algorithms, and again form the morphisms of a monoidal category. In this paper, we show that the two concepts are tightly linked. We show both that there is a faithful, identity-on-objects symmetric monoidal functor embedding a category of asymmetric lenses into the category of learners, and furthermore there is such a functor embedding the category of learners into a category of symmetric lenses.