From Abelianization to Tangent Categories
Sacha Ikonicoff, Jean-Simon Pacaud Lemay, Tim Van der Linden
TL;DR
This work develops a universal method to construct tangent categories from linear assignments, revealing that abelianization provides a natural tangent structure on groups via $\mathcal{T}(G)=G\times\mathrm{Ab}(G)$. It establishes a bidirectional correspondence between linear assignments and linear projectors, and shows how linear algebras, differential objects, and differential bundles arise from these assignments, with $\mathsf{GRP}$ furnishing concrete Rosický tangent structure through abelianization. The framework extends to monadic linear assignments and linear reflectors, notably recovering abelianization as a reflector and linking to Eilenberg–Moore theory. Generalizing further to unital regular categories, the authors obtain a wide array of new tangent categories where differential bundles correspond to commutative monoids, giving rise to examples across algebraic varieties such as monoids, Jónsson–Tarski varieties, Lie algebras, crossed modules, and Hopf algebras. Overall, the paper significantly broadens the landscape of tangent categories beyond geometry, tying tangent calculus to a rich algebraic and categorical structure via abelianization and its generalizations.
Abstract
A tangent category is a category with an endofunctor, called the tangent bundle functor, which is equipped with various natural transformations that capture essential properties of the classical tangent bundle of smooth manifolds. In this paper, we show that, surprisingly, the category of groups is a tangent category whose tangent bundle functor is induced by abelianization and whose differential bundles correspond to abelian groups. We generalize this construction by introducing the concept of linear assignments, which are endofunctors assigning to every object a commutative monoid in a natural and idempotent manner. We then show that a linear assignment induces a tangent bundle functor, whose differential bundles correspond to a notion of linear algebras. We show that any finitely cocomplete regular unital category is a tangent category whose tangent bundle functor is induced by the canonical abelianization functor, which is a monadic linear assignment. This allows us to provide multiple new examples of tangent categories including monoids, pointed magmas, loops, non-unital rings, Jónsson--Tarski varieties, and pointed Mal'tsev varieties.