A functorial approach to differential calculus

Abstract: We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets). To prepare this approach, we define a new, symmetric, presentation of differential calculus, whose main feature is the central r{^o}le played by the anchor map, which we study in detail. Our aim for developing this theory is twofold: (1) define a setting for calculus over any commutative ring, including finite rings; (2) define a setting that can be generalized to categories of graded rings (super differential calculus).