Quasitopoi, logic and Heyting-valued models
type: Item
instance of: scholarly article
title: Quasitopoi, logic and Heyting-valued models
zbMATH Open document ID: 0597.18001
DOI: 10.1016/0022-4049(86)90077-0
author: G. P. Monro
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: This paper deals with various facets of quasitopoi, which are relevant to the interpretation and modelling of first-order logic in quasitopoi. The first section deals with topologies. A topology \(\sigma\) on a quasitopos Q is strong iff the \(\sigma\)-closure of every mono is a strong mono. If \(\sigma\) is strong, the \(\sigma\)-sheaves, \(sh_{\sigma}(Q)\), form a topos. The strong topology which assigns to every mono the strong monic part of its epic/strong monic factorization has as its sheaves what the author calls the coarse objects of the quasitopos Q. A quasitopos is called solid iff it has disjoint coproducts. Solid quasitopoi behave well with respect to several topos theoretic constructions. In particular, if Q is solid and \(X\in Q\), then Q/X is again solid and if \({\mathcal C}\) is an internal category in Q, then \(Q^{{\mathcal C}}\) is a solid quasitopos. Next, the author describes the interpretation of first order logic in quasitopoi. It is analogous to the case of topoi, except that predicates must be interpreted as strong monos rather than arbitrary monos. Finally, if H is a complete Heyting algebra, a solid quasitopos of models, Mod(H), is described. The topos of coarse objects of Mod(H) turns out to be precisely sh(H).
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18A15
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 03C07
zbMATH DE Number: 3961916
zbMATH Keywords: (E,M)-factorization
zbMATH Keywords: quasitopoi
zbMATH Keywords: first-order logic
zbMATH Keywords: \(\sigma \) -sheaves
zbMATH Keywords: topos
zbMATH Keywords: strong topology
zbMATH Keywords: solid quasitopos
zbMATH Keywords: complete Heyting algebra
zbMATH Keywords: coarse objects
reviewed by: K. I. Rosenthal
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90077-0
OpenAlex ID: W2071604874
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description: scientific article; zbMATH DE number 3961916