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 Adámek's construction of an initial algebra for $(-)\mathbf{Cat}$ does not terminate; more precisely, we show that Adámake's construction of an initial algebra for the endofunctor $(-){\mathbf{Cat}}^{<\lambda }$ that sends a symmetric monoidal $(\infty ,1)$-category $\mathcal{V}$ to the $(\infty ,1)$-category of $\mathcal{V}$-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.

6 pages