This paper studies the homotopy theory of parametrized spectrum objects in a model category from a global point of view. More precisely, for a model category $M$ satisfying suitable conditions, we construct a map of model categories $TM \to M$, called the tangent bundle, whose fiber over an object in $M$ is a model category for spectra in its over-category. We show that the tangent bundle is a relative model category and presents the $\infty$-categorical tangent bundle, as constructed by Lurie. Moreover, the tangent bundle $TM$ inherits an enriched model structure from $M$. This additional structure is used in subsequent work to identify the tangent bundles of algebras over an operad and of enriched categories, but may be of independent interest.
Keywords: Tangent category, model category, model fibration, spectrum
2010 MSC: 55P42, 18G55, 18D30
Theory and Applications of Categories, Vol. 34, 2019, No. 33, pp 1039-1072.