Classifying Infinity Topoi via Weighted Limits

We construct classifying $\infty$-topoi by showing that the $(\infty,2)$-category of topoi has weighted limits. We show that several prestacks of interest have a classifying topos, including the prestack of spectra.

💡 Research Summary

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The paper “Classifying Infinity Topoi via Weighted Limits” develops a new categorical framework for constructing classifying ∞‑topoi by showing that the (∞,2)-category of topoi admits weighted limits. The authors, Di Liberti and Meadows, pursue a concrete, logic‑free approach inspired by Johnstone’s classical construction of classifying topoi, but lifted to the higher‑categorical setting.

The first major technical achievement is Theorem 5.1, which asserts that the (∞,2)-category TOP B of bounded topoi and bounded geometric morphisms possesses all weighted limits. To reach this result, the paper proves a higher‑categorical analogue of a classical 2‑categorical theorem: an (∞,2)-category has weighted limits iff it has ordinary (conical) limits and cotensors with the arrow ∆¹. This is formalized in Theorem 2.7. The proof relies on Lurie’s completeness results for ∞‑categories (the (∞,1)-category of topoi has all limits) and on the model structure for scaled simplicial sets developed by Gagna–Harvey–Lurie (GHL). The authors show that cotensoring with ∆¹ suffices to generate the “arrow powers” needed to weight diagrams, thereby establishing the existence of all weighted limits in TOP B.

Having secured weighted limits, the authors turn to the construction of classifying ∞‑topoi. They define a prestack (T : \mathrm{TOP}^{op} \to \mathrm{CAT}_\infty) as a rule assigning to each ∞‑topos (\mathcal{E}) an ∞‑category of “models of a theory” inside (\mathcal{E}). If such a prestack is representable by an object (B