Projects per year

### 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 Jónsson 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.

Original language | English |
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Article number | 102831 |

Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Annals of Pure and Applied Logic |

Volume | 171 |

Issue number | 10 |

DOIs | |

Publication status | Published - Dec 2020 |

### Keywords

- Ultrafilter
- Ultracategory
- Canonical extension
- Topos of types
- Locally connected classifying topos

## Fingerprint Dive into the research topics of 'Ultrafilters, finite coproducts and locally connected classifying toposes'. Together they form a unique fingerprint.

## Projects

## Working synthetically in higher categorical structures

Lack, S., Verity, D., Garner, R. & Street, R.

19/06/19 → 18/06/22

Project: Other

## Monoidal categories and beyond: new contexts and new applications

Street, R., Verity, D., Lack, S., Garner, R. & MQRES Inter Tuition Fee only, M. I. T. F. O.

30/06/16 → 17/06/19

Project: Research