Parametrized higher category theory and higher algebra: Exposé I -- Elements of parametrized higher category theory

Abstract

We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we are able at this stage. Simple operations, such as the formation of opposites and the formation of functor $\infty$-categories, become slightly more involved in the parametrized setting, but we explain precisely how to perform these constructions. All of these constructions can be performed explicitly, without resorting to such acts of desperation as straightening. The key results of this Exposé are: (1) a universal characterization of the $T$-$\infty$-category of $T$-objects in any $\infty$-category, (2) the existence of an internal Hom for $T$-$\infty$-categories, and (3) a parametrized Yoneda lemma.

arXiv comment23 pages

References in corpus (3)

• On the fibrewise effective Burnside $\infty$-category
• Fibrations in $\infty$-category theory
• Equivalence of models for equivariant $(\infty ,1)$-categories

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• On categories of slices
• On the parametrized Tate construction
• Yoneda's lemma for internal higher categories
• On the construction of limits and colimits in $\infty$-categories