Parametrized higher category theory

Abstract

We develop foundations for the category theory of $\infty$-categories parametrized by a base $\infty$-category. Our main contribution is a theory of indexed homotopy limits and colimits, which specializes to a theory of $G$-colimits for $G$ a finite group when the base is chosen to be the orbit category of $G$. We apply this theory to show that the $G$-$\infty$-category of $G$-spaces is freely generated under $G$-colimits by the contractible $G$-space, thereby affirming a conjecture of Mike Hill.

83 pages, to appear in Algebraic & Geometric Topology

References in corpus (3)

• Higher Topos Theory
• Parametrized higher category theory and higher algebra: Exposé I -- Elements of parametrized higher category theory
• A user's guide to co/cartesian fibrations

Cited by in corpus (12)

• Global homotopy theory via partially lax limits
• Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra
• Reedy Model Structures in Families
• On the parametrized Tate construction
• An Extension of the Kazhdan-Lusztig Equivalence
• Parametrized stability and the universal property of global spectra
• Parametrized higher category theory II: Universal constructions
• Recollements and stratification
• Partial parametrized presentability and the universal property of equivariant spectra
• A local-global principle for parametrized $\infty$-categories
• The Adams isomorphism revisited
• Applications of equivariant factorization homology