# Learning in Infinitesimal Non-Compositional Sketches

arXiv: https://arxiv.org/abs/2607.15107 · HTML: https://mlsanity.com/paper/2607.15107/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 33 references that are in the map.
- First posted on arXiv 16 Jul 2026.

## Builds on (most cited first, 7 of 7)

- [Backprop as Functor: A compositional perspective on supervised learning](https://mlsanity.com/paper/1711.10455/) (2017)
- [Automated Crossword Solving](https://mlsanity.com/paper/2205.09665/) (2022)
- [Latent Confounded Causal Discovery via Lie Bracket Geometry](https://mlsanity.com/paper/2606.19610/) (2026)
- [Kan Extension Transformers: A Categorical Unification of Attention, Diffusion, and Predict-Detach Self-Conditioning](https://mlsanity.com/paper/2605.27259/) (2026)
- [Infinitesimal Causality](https://mlsanity.com/paper/2606.24621/) (2026)
- [Universal Decision Learners](https://mlsanity.com/paper/2605.30694/) (2026)
- [Agentic Skill Optimization over Lie Algebroids](https://mlsanity.com/paper/2607.11493/) (2026)

## Built on by (most cited first, 1 of 1)

- [When the Canonical Completion Is Wrong: Formalizing and Measuring the Jump in Large Language Models](https://mlsanity.com/paper/2608.26187/) (2026)

## Abstract (arXiv)

This paper develops a categorical framework -- Learning in Infinitesimal Non-Compositional Sketches (LINCS) -- as the repair of non-compositionality: failures of diagrams to factor through quotient sketches lifted to the tangent category setting. Machine learning problems are specified as sketches: graphs with commutativity conditions mathcal D, limit cones mathcal L, and colimit cocones mathcal K, generalizing the usual scalarization of loss functions or vector space assumptions. Non-compositionality is defined purely as failure of a universal factorization problem, not as arithmetic error between the desired and actual predictions. Given a learning sketch mathbb S=(S,mathcal D,mathcal L,mathcal K), whose underlying graph is S, and a model D:J rightarrow C, the base defect is the obstruction to factorization mboxObs(mboxFactmathbb S(D)). The tangent lift applies the tangent functor T to obtain TD:J rightarrow C, and LINCS is defined as the obstruction mboxObs(mboxFactmathbb S(TD)) -- asking whether infinitesimal perturbations preserve the compositionality constraints.The paper also introduces Tangent Learning Sketches, which are sketches equipped with Cockett-Cruttwell tangent structure. The paper defines the INC endofunctor, which iterates the tangent lift, producing a tower D,TD,T2D, cdots of factorization problems. ML is thereby formulated as the search for a coalgebraic fixed point where successive tangent unfoldings stabilize (nu T_mboxINC). Using the Aczel--Mendler theorem, we prove existence of a final INC coalgebra whenever T_mboxINC admits a set-based class realization that creates its final carrier. A detailed experimental evaluation of LINCS is underway in a number of concrete ML settings, including deep learning, large language models, and reinforcement learning, and is described in companion papers.