Learning in Infinitesimal Non-Compositional Sketches

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 $D:J\to C$, the base defect is the obstruction to factorization $Obs(Fac{t}_{\mathbb{S}}(D))$. The tangent lift applies the tangent functor $T$ to obtain $TD:J\to C$, and LINCS is defined as the obstruction $Obs(Fac{t}_{\mathbb{S}}(TD))$ – 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 $D,TD,T^{2}D,\cdots$ of factorization problems. ML is thereby formulated as the search for a coalgebraic fixed point where successive tangent unfoldings stabilize ($\nu T_{INC}$). Using the Aczel–Mendler theorem, we prove existence of a final INC coalgebra whenever $T_{INC}$ 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.