  Tangent categories provide a categorical axiomatization of the tangent
bundle. There are many interesting examples and applications of tangent
categories in a variety of areas such as differential geometry, algebraic
geometry, algebra, and even computer science. The purpose of this paper is to
expand the theory of tangent categories in a new direction: the theory of
operads. The main result of this paper is that both the category of algebras of
an operad and its opposite category are tangent categories. The tangent bundle
for the category of algebras is given by the semi-direct product, while the
tangent bundle for the opposite category of algebras is constructed using the
module of K\"ahler differentials, and these tangent bundles are in fact
adjoints of one another. To prove these results, we first prove that the
category of algebras of a coCartesian differential monad is a tangent category.
We then show that the monad associated to any operad is a coCartesian
differential monad. This also implies that we can construct Cartesian
differential categories from operads. Therefore, operads provide a bountiful
source of examples of tangent categories and Cartesian differential categories,
which both recaptures previously known examples and also yield new interesting
examples. We also discuss how certain basic tangent category notions recapture
well-known concepts in the theory of operads.
