Canonical differential calculi via functorial geometrization

Abstract

Given a category $\mathcal{E}$, we establish sufficient conditions on a faithful isofibration $\mathcal{E}\rightarrow\operatorname{Mon}(\mathcal{V})$ valued in the category of monoids internal to a monoidal additive category $\mathcal{V}$ such that $\mathcal{E}$ admits a canonical functor to the category of first order differential calculi in $\mathcal{V}$. Generalizing the procedure of extending a first order differential calculus to its maximal prolongation to this setting, we obtain a canonical functor from $\mathcal{E}$ to the category of differential calculi in $\mathcal{V}$. This yields a simultaneous generalization of the de Rham complex on $C^{\infty}$-rings, the K\"{a}hler differentials on commutative algebras, and the universal differential calculus on associative algebras. As a consequence, such categories $\mathcal{E}$ admit natural analogues of the notions of smooth map and diffeomorphism, as well as a functorial de Rham theory. Moreover, whenever two such faithful isofibrations to $\operatorname{Mon}(\mathcal{V})$ factor suitably, their corresponding de Rham functors are related via a comparison map. Developing this theory requires first extending the noncommutative geometry formalism of differential calculi from associative algebras to the setting of monoids internal to monoidal additive categories.