scholarly journals Representing Geometric Morphisms Using Power Locale Monads

2011 ◽  
Vol 21 (1) ◽  
pp. 15-47
Author(s):  
Christopher F. Townsend
Keyword(s):  
Geometric Morphisms

Flat functors and classifying toposes

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
General Theory ◽  
Geometric Theory ◽  
Small Category ◽  
Independent Interest ◽  
Yoneda Lemma ◽  
Geometric Morphisms ◽  
The Way

This chapter develops a general theory of extensions of flat functors along geometric morphisms of toposes; the attention is focused in particular on geometric morphisms between presheaf toposes induced by embeddings of categories and on geometric morphisms to the classifying topos of a geometric theory induced by a small category of set-based models of the latter. A number of general results of independent interest are established on the way, including developments on colimits of internal diagrams in toposes and a way of representing flat functors by using a suitable internalized version of the Yoneda lemma. These general results will be instrumental for establishing in Chapter 6 the main theorem characterizing the class of geometric theories classified by a presheaf topos and for applying it.

Simply Connected Limits

1990 ◽  
Vol 42 (4) ◽  
pp. 731-746 ◽  
Author(s):  
Robert Paré
Keyword(s):  
Left Adjoint ◽  
Topos Theory ◽  
Essential Ingredient ◽  
Original Definition ◽  
The Subject ◽  
Definition Of ◽  
Simply Connected ◽  
Geometric Morphisms ◽  
Exact Categories

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

scholarly journals Syntax and Semantics of the logic L_omega omega^lambda

1997 ◽  
Vol 4 (22) ◽  
Author(s):  
Carsten Butz
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Generic Model ◽  
First Order ◽  
Geometric Morphisms

In this paper we study the logic L_omega omega^lambda , which is first order logic<br /> extended by quantification over functions (but not over relations).<br > We give the syntax of the logic, as well as the semantics in Heyting<br /> categories with exponentials. Embedding the generic model of a theory<br /> into a Grothendieck topos yields completeness of L_omega omega^lambda with respect<br /> to models in Grothendieck toposes, which can be sharpened to completeness<br />with respect to Heyting valued models. The logic L_omega omega^lambda is the<br />strongest for which Heyting valued completeness is known. Finally,<br />we relate the logic to locally connected geometric morphisms between toposes.

Geometric Morphisms

1994 ◽  
pp. 347-420
Author(s):  
Saunders Mac Lane ◽  
Ieke Moerdijk
Keyword(s):  
Geometric Morphisms

Essential geometric morphisms between toposes over finite sets

1980 ◽  
Vol 87 (1) ◽  
pp. 21-24
Author(s):  
John Haigh
Keyword(s):  
Finite Sets ◽  
Generating Set ◽  
Cogenerating Set ◽  
Geometric Morphism ◽  
Geometric Morphisms

We show that if {Gi}J ε I is a generating set for an (elementary) topos ℰ then {P(Gi)}iεI is a cogenerating set for x2130;. From this we show that if topos ℰ contains an object G whose subobjects generate ℰ, then ΩG is a cogenerator for ℰ. Let denote the topos of finite sets and functions. We also show that if ℰ1 is a topos and ℰ2 is a bounded -topos then every geometric morphism ℰ1 → ℰ2 is essential.

scholarly journals Points in topoi of sheaves over distributive lattices

1979 ◽  
Vol 20 (2) ◽  
pp. 273-279
Author(s):  
M. Adelman
Keyword(s):  
Topological Space ◽  
Distributive Lattices ◽  
Irreducible Components ◽  
Geometric Morphisms

This article generalizes the well known theorem that the geometric morphisms from the category of sets to a category of set-valued sheaves on a topological space correspond to the irreducible components of the topological space. As irreducible components are not available in any topos more general than a spatial one, they are characterized in terms of filters of open sets - which are available in any topos. It is then seen that the theorem phrased in these terms generalizes to sheaves on any lattice with reasonable distributivity conditions.

scholarly journals Strongly algebraic = SFP (topically)

2001 ◽  
Vol 11 (6) ◽  
pp. 717-742 ◽  
Author(s):  
STEVEN VICKERS
Keyword(s):  
Information Systems ◽  
Geometric Theory ◽  
Upper Bounds ◽  
Finite Model ◽  
Finite Structure ◽  
Algebraic Domains ◽  
Finite Posets ◽  
Geometric Morphisms

Certain ‘Finite Structure Conditions’ on a geometric theory are shown to be sufficient for its classifying topos to be a presheaf topos. The conditions assert that every homomorphism from a finite structure of the theory to a model factors via a finite model, and they hold in cases where the finitely presentable models are all finite.The conditions are shown to hold for the theory of strongly algebraic (or SFP) information systems and some variants, as well as for some other theories already known to be classified by presheaf toposes.The work adheres to geometric constructivism throughout, and in consequence provides ‘topical’ categories of domains (internal in the category of toposes and geometric morphisms) with an analogue of Plotkin's double characterization of strongly algebraic domains, by sets of minimal upper bounds and by sequences of finite posets.

scholarly journals Triposes as a generalization of localic geometric morphisms

2021 ◽  
pp. 1-10
Author(s):  
Jonas Frey ◽  
Thomas Streicher
Keyword(s):  
Heyting Algebras ◽  
The One ◽  
Phd Thesis ◽  
Geometric Morphisms

Abstract In Hyland et al. (1980), Hyland, Johnstone and Pitts introduced the notion of tripos for the purpose of organizing the construction of realizability toposes in a way that generalizes the construction of localic toposes from complete Heyting algebras. In Pitts (2002), one finds a generalization of this notion eliminating an unnecessary assumption of Hyland et al. (1980). The aim of this paper is to characterize triposes over a base topos ${\cal S}$ in terms of so-called constant objects functors from ${\cal S}$ to some elementary topos. Our characterization is slightly different from the one in Pitts’s PhD Thesis (Pitts, 1981) and motivated by the fibered view of geometric morphisms as described in Streicher (2020). In particular, we discuss the question whether triposes over Set giving rise to equivalent toposes are already equivalent as triposes.

Sign in / Sign up

Export Citation Format

Share Document