  This paper studies the homotopy theory of parameterized spectrum objects in a
model category from a global point of view. More precisely, for a model
category $\mathcal{M}$ satisfying suitable conditions, we construct a relative
model category $\mathcal{TM} \to \mathcal{M}$, called the tangent bundle, whose
fibers are models for spectra in the various over-categories of $\mathcal{M}$,
and which presents the $\infty$-categorical tangent bundle. Moreover, the
tangent bundle $\mathcal{TM}$ inherits an enriched model structure when such a
structure exists on $\mathcal{M}$. This additional structure is used in
subsequent work to identify the tangent bundles of algebras over an operad and
of enriched categories.
