Ultrafilters, finite coproducts and locally connected classifying toposes

TL;DR

This paper unifies various ultrafilter concepts through a category-theoretic framework, revealing new insights into model theory and topos theory, including a concrete description of the locally connected classifying topos for first-order theories.

Contribution

It establishes a fundamental equivalence between finite-coproduct-preserving endofunctors of Set and presheaves on ultrafilters, connecting ultrafilter theory with topos-theoretic structures.

Findings

Unified ultrafilter concepts via category theory

Revealed the model-theoretic relation between types and elements

Provided a concrete description of the classifying topos for first-order theories

Abstract

We prove a single category-theoretic result encapsulating the notions of ultrafilters, ultrapower, ultraproduct, tensor product of ultrafilters, the Rudin--Kiesler partial ordering on ultrafilters, and Blass's category of ultrafilters UF. The result in its most basic form states that the category FC(Set,Set) of finite-coproduct-preserving endofunctors of Set is equivalent to the presheaf category [UF,Set]. Using this result, and some of its evident generalisations, we re-find in a natural manner the important model-theoretic realisation relation between n-types and n-tuples of model elements; and draw connections with Makkai and Lurie's work on conceptual completeness for first-order logic via ultracategories. As a further application of our main result, we use it to describe a first-order analogue of J\'onsson and Tarski's canonical extension. Canonical extension is an algebraic formulation of the link between Lindenbaum--Tarski and Kripke semantics for intuitionistic and modal logic, and extending it to first-order logic has precedent in the topos of types construction studied by Joyal, Reyes, Makkai, Pitts, Coumans and others. Here, we study the closely related, but distinct, construction of the locally connected classifying topos of a first-order theory. The existence of this is known from work of Funk, but the description is inexplicit; ours, by contrast, is quite concrete.