arXiv subjects

Zach Goldthorpe

Publications and source records attributed to Zach Goldthorpe.

3 recordsLinked to original sources

arXiv2024

Sheaves of $(\infty, \infty)$-categories

We provide a functorial presentation of the $(\infty, 1)$-category of sheaves of $(n, r)$-categories for all $-2 \leq n\leq\infty$ and $0 \leq r\leq n+2$ based on complete Segal space objects. In this definition, the equivalences of sheaves of $(\infty, \infty)$-categories are defined inductively, so we also provide a localisation at the coinductive equivalences to define the $(\infty, 1)$-category of sheaves of $ω$-categories as well. We prove that the $(\infty, 1)$-category of sheaves of $(\infty, \infty)$-categories and the $(\infty, 1)$-category of sheaves of $ω$-categories both define distributors over the underlying topos of sheaves of spaces. Moreover, we show that these distributors define terminal and initial fixed points with respect to the construction of complete Segal space objects in a distributor. We conclude with a sheafification result: the category of sheaves of $(n, r)$-categories over a site $\mathscr{C}$ can be presented as a strongly reflective localisation of $\mathbf{Fun}(\mathscr{C}^{\mathrm{op}}, \mathbf{Cat}_{(n, r)})$, where the localisation functor preserves fibre products over $\mathbf{Sh}(\mathscr{C})$, with the analogous result also holding for sheaves of $ω$-categories.

math.CT↗

arXiv2024

Homotopy theories of $(\infty, \infty)$-categories as universal fixed points with respect to enrichment

We show that both the $\infty$-category of $(\infty, \infty)$-categories with inductively defined equivalences, and with coinductively defined equivalences, satisfy universal properties with respect to weak enrichment in the sense of Gepner and Haugseng. In particular, we prove that $(\infty, \infty)$-categories with coinductive equivalences form a terminal object in the $\infty$-category of fixed points for enrichment, and that $(\infty, \infty)$-categories with inductive equivalences form an initial object in the subcategory of locally presentable fixed points. To do so, we develop an analogue of Adámek's construction of free endofunctor algebras in the $\infty$-categorical setting. We prove that $(\infty, \infty)$-categories with coinductive equivalences form a terminal coalgebra with respect to weak enrichment, and $(\infty, \infty)$-categories with inductive equivalences form an initial algebra with respect to weak enrichment.

math.CT↗

arXiv2024

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

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.

math.CT↗