On the fibrewise effective Burnside $\mathrm{\infty }$-category

Effective Burnside $\mathrm{\infty }$-categories are the centerpiece of the $\mathrm{\infty }$-categorical approach to equivariant stable homotopy theory. In this 'etude, we recall the construction of the twisted arrow $\mathrm{\infty }$-category, and we give a new proof that it is an $\mathrm{\infty }$-category, using an extremely helpful modification of an argument due to Joyal–Tierney. The twisted arrow $\mathrm{\infty }$-category is in turn used to construct the effective Burnside $\mathrm{\infty }$-category. We employ a variation on this theme to construct a fibrewise effective Burnside $\mathrm{\infty }$-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.