Classifying Infinity Topoi via Weighted Limits

TL;DR

The paper develops a concrete, topos-theoretic framework for classifying ∞-topoi by proving the (∞,2)-category TOP_B has weighted limits, enabling the construction of classifying topoi for a broad class of theories (including spectra) via geometric sketches. It combines weighted-limit techniques with internal higher category theory in ∞-topoi and a key flat-functor lemma to establish the object classifier and a representability paradigm for prestack theories. The results provide a robust, logic-agnostic pathway to classify semantic models across topoi, unifying Lawvere theories, truncated/object theories, spectra, and sketches under the classifying topos umbrella. This framework promises applications in synthetic algebraic geometry and higher topos theory by enabling compact, universal classifiers for a wide range of higher-categorical constructions.

Abstract

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.

Theorems & Definitions (79)

• Theorem 1: \ref{['thm6.1']}
• Example 2: Models fo a Lawvere theory
• Example 3: \ref{['spectra']}
• Theorem 4: \ref{['classtoposforgeoske']}
• Definition 1.6
• Theorem 1.7
• Definition 1.8
• Definition 1.10
• Definition 1.11
• Theorem 1.12