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 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.