Tangent categories as a bridge between differential geometry and algebraic geometry

Abstract

Discussions of tangent vectors, tangent spaces, and differentials are important in both differential geometry and algebraic geometry. In this paper, we use the abstract notion of a tangent category to make some of these commonalities precise. In particular, we focus on the idea of a differential bundle in a tangent category, which gives a new way to compare smooth vector bundles and modules. The results of this paper also give a new characterization of the opposite category of modules over a commutative ring and the opposite category of quasicoherent sheaves.

The Tangent Categories of Algebras over an Operad

“…The main objective of this paper is to show that both ALGP and the opposite category ALG op P are tangent categories (Theorem 4.3.3 and Theorem 4.4.4), whose tangent bundles are adjoints to one another (Lemma 4.4.2). This is a generalization of the fact that ALGCom and ALG op Com are both well-known examples of tangent categories, see for example [11] for full details. Briefly, the tangent bundle of a commutative R-algebra A in ALGCom is given by the algebra of dual numbers over A (Example 4.3.5):…”

“…The tangent category ALG op Com is closely related to algebraic geometry, as explained in [11]. Indeed, it is famously known that ALG op Com , the opposite category of commutative algebras, is equivalent to the category of affine schemes over R, the building blocks in algebraic geometry.…”

“…Therefore, the tangent bundles T and T • are mutual adjoints, as desired. In particular, following the discussions of [11], we may interpret ALG op P as a tangent category model of algebraic geometry relative to the operad P. It is worth mentioning that, while the tangent bundle T is mostly the same for each operad, the adjoint tangent bundle T • can vary quite drastically from operad to operad.…”

“…In particular, we will discuss the adjoint tangent structure of algebras, commutative algebras, and Lie algebras. There are also important examples of tangent categories with adjoint tangent structures related to synthetic differential geometry [15], differential linear logic [10], and algebraic geometry [11].…”

“…Indeed, the canonical example of a tangent category is the category of smooth manifolds, where the tangent structure is induced by the classical tangent bundle of a smooth manifold. Recently, there has been an upswing on new interesting examples and novel applications of tangent categories beyond differential geometry such as in commutative algebra and algebraic geometry [11], and even in computer science, in particular in relation to differential linear logic [10] and the differential lambda calculus [29]. The objective of this paper is to further expand the theory and applications of tangent categories into a new direction: the theory of operads.…”