Parametrized higher category theory

TL;DR

This paper develops a foundational theory for parametrized $mbda$-categories, introducing indexed homotopy limits and colimits, and applies it to $G$-spaces, confirming a conjecture of Mike Hill.

Contribution

It introduces a new framework for parametrized higher category theory and develops a theory of indexed homotopy limits and colimits, including $G$-colimits, with applications to $G$-spaces.

Findings

The $G$-$mbda$-category of $G$-spaces is generated by the contractible $G$-space under $G$-colimits.

The theory specializes to recover $G$-colimits for finite groups.

Confirmed a conjecture of Mike Hill regarding $G$-spaces.

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.