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

Zach Goldthorpe

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 $(-)\mathbf{Cat}^{<λ}$ that sends a symmetric monoidal $(\infty, 1)$-category $\mathscr{V}$ to the $(\infty, 1)$-category of $\mathscr{V}$-enriched categories with at most $λ$ equivalence classes of objects terminates in precisely $λ$ 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.

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 over general symmetric monoidal -categories. We prove that Adámek's construction of an initial algebra for does not terminate; more precisely, we show that Adámake's construction of an initial algebra for the endofunctor that sends a symmetric monoidal -category to the -category of -enriched categories with at most equivalence classes of objects terminates in precisely steps. We also prove that an initial algebra for the endofunctor exists nonetheless, and characterise it as the -category consisting of those -categories that satisfy a weak finiteness property we call Noetherian.
Paper Structure (3 sections, 7 theorems, 10 equations)

This paper contains 3 sections, 7 theorems, 10 equations.

Table of Contents

  1. Preliminaries
  2. Measuring finiteness
  3. Large convergence

Key Result

Lemma 4

The $\infty$-categories $\mathbf{Cat}_{<\theta}$ can be constructed through transfinite induction via enrichment:

Theorems & Definitions (20)

  • Definition 1
  • Remark 2
  • Example 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • ...and 10 more