On the fibrewise effective Burnside $\infty$-category

TL;DR

This paper develops a fibrewise version of the effective Burnside $$-category within the context of equivariant homotopy theory, introducing marbled simplicial sets and refining proof techniques for $$-categories.

Contribution

It constructs a fibrewise effective Burnside $$-category and introduces marbled simplicial sets, providing new methods and proofs in the theory of $$-categories.

Findings

Successful construction of fibrewise effective Burnside $$-category

Introduction of marbled simplicial sets for $$-category theory

Refined proof techniques based on Joyal--Tierney argument

Abstract

Effective Burnside $\infty$-categories are the centerpiece of the $\infty$-categorical approach to equivariant stable homotopy theory. In this \'etude, we recall the construction of the twisted arrow $\infty$-category, and we give a new proof that it is an $\infty$-category, using an extremely helpful modification of an argument due to Joyal--Tierney. The twisted arrow $\infty$-category is in turn used to construct the effective Burnside $\infty$-category. We employ a variation on this theme to construct a fibrewise effective Burnside $\infty$-category. To show that this constuctionworks fibrewise, we introduce a fragment of a theory of what we call marbled simplicial sets, and we use a yet further modified form of the Joyal--Tierney argument.