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The tangent bundle of a model category

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Yonatan Harpaz, Joost Nuiten, Matan Prasma

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.

Published 2019-10-18.

http://www.tac.mta.ca/tac/volumes/34/33/34-33.pdf

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