Semisimplicity manifesting as categorical smallness

Abstract

For a compact group $\mathbb{G}$, the functor from unital Banach algebras with contractive morphisms to metric spaces with 1-Lipschitz maps sending a Banach algebra $A$ to the space of $\mathbb{G}$-representations in $A$ preserves filtered colimits. Along with this, we prove a number of analogues: one can substitute unitary representations in $C^*$-algebras, as well as semisimple finite-dimensional Banach algebras (or finite-dimensional $C^*$-algebras) for $\mathbb{G}$. These all mimic results on the metric-enriched finite generation/presentability of finite-dimensional Banach spaces due to Ad{\'a}mek and Rosick{\'y}. We also give an alternative proof of that finite presentability result, along with parallel results on functors represented by compact metric, metric convex, or metric absolutely convex spaces.