A note on Noetherian $(\infty, \infty)$-categories
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911808806191104 |
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| author | Goldthorpe, Zach |
| author_facet | Goldthorpe, Zach |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14827 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on Noetherian $(\infty, \infty)$-categories Goldthorpe, Zach Category Theory 18D20, 19D23, 18C10, 18N65 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. |
| title | A note on Noetherian $(\infty, \infty)$-categories |
| topic | Category Theory 18D20, 19D23, 18C10, 18N65 |
| url | https://arxiv.org/abs/2403.14827 |