A note on Noetherian $(\infty, \infty)$-categories

Abstract

The purpose of this note is to resolve a conjecture in arXiv:2307.00442(4), regarding the initial algebra for the enrichment endofunctor $(-)\mathbf{Cat}$ over general symmetric monoidal $(\infty, 1)$-categories. We prove that Ad\'amek's construction of an initial algebra for $(-)\mathbf{Cat}$ does not terminate; more precisely, we show that Ad\'amake's construction of an initial algebra for the endofunctor $(-)\mathbf{Cat}^{<\lambda}$ that sends a symmetric monoidal $(\infty, 1)$-category $\mathscr{V}$ to the $(\infty, 1)$-category of $\mathscr{V}$-enriched categories with at most $\lambda$ equivalence classes of objects terminates in precisely $\lambda$ steps. We also prove that an initial algebra for the endofunctor $(-)\mathbf{Cat}$ exists nonetheless, and characterise it as the $(\infty, 1)$-category consisting of those $(\infty, \infty)$-categories that satisfy a weak finiteness property we call Noetherian.