  Cartesian differential categories provide a categorical framework for
multivariable differential calculus and also the categorical semantics of the
differential $\lambda$-calculus. Taylor series expansion is an important
concept for both differential calculus and the differential $\lambda$-calculus.
In differential calculus, a function is equal to its Taylor series if its
sequence of Taylor polynomials converges to the function in the analytic sense.
On the other hand, for the differential $\lambda$-calculus, one works in a
setting with an appropriate notion of algebraic infinite sums to formalize
Taylor series expansion. In this paper, we provide a formal theory of Taylor
series in an arbitrary Cartesian differential category without the need for
converging limits or infinite sums. We begin by developing the notion of Taylor
polynomials of maps in a Cartesian differential category and then show how
comparing Taylor polynomials of maps induces an ultrapseudometric on the
homsets. We say that a Cartesian differential category is Taylor if maps are
entirely determined by their Taylor polynomials. The main results of this paper
are that in a Taylor Cartesian differential category, the induced
ultrapseudometrics are ultrametrics and that for every map $f$, its Taylor
series converges to $f$ with respect to this ultrametric. This framework
recaptures both Taylor series expansion in differential calculus via analytic
methods and in categorical models of the differential $\lambda$-calculus (or
Differential Linear Logic) via infinite sums.
