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Abstract
We prove a single categorytheoretic 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 finitecoproductpreserving endofunctors of Set is equivalent to the presheaf category [UF,Set]. Using this result, and some of its evident generalisations, we refind in a natural manner the important modeltheoretic realisation relation between ntypes and ntuples of model elements; and draw connections with Makkai and Lurie's work on conceptual completeness for firstorder logic via ultracategories.
As a further application of our main result, we use it to describe a firstorder 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 firstorder 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 firstorder 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 

Article number  102831 
Pages (fromto)  129 
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
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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