## Abstract (2010)

### BibTeX

@MISC{Awodey10abstract,

author = {Steve Awodey and Henrik Forssell},

title = {Abstract},

year = {2010}

}

### OpenURL

### Abstract

From alogical pointof view, Stoneduality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2. In the present work, we generalize the entire arrangement from propositional to first-order logic. Boolean algebras are replaced by Booleancategories presentedbytheoriesinfirst-orderlogic, andspaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed first in the form of a contravariant adjunction, is established by homming into a common dualizing object, now Sets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology.

### Citations

203 |
Locally presentable and accessible categories
- Adámek, Rosický
- 1994
(Show Context)
Citation Context ...certain condition. Now, the category ModT of models and homomorphisms of a Cartesian theory T has limits and filtered colimits (but not, in general, regular epis), and Gabriel-Ulmer duality (see e.g. =-=[5]-=-) informs us, among other things, that the definable set functors for Cartesian formulas (relative to T) can be characterized as the limit and filtered colimit preserving functors ModT → Sets (and tha... |

114 |
An extension of the Galois theory of Grothendieck
- Joyal, Tierney
- 1984
(Show Context)
Citation Context ...rem 1.6.11, following the lines of [10], is given in [9, Chapter 3]. It proceeds by showing that the spatial covering m : Sh(XT) �� Sh(CT) of Section 1.2 is an open surjection and thus, by results of =-=[13]-=-, an effective descent morphism. The groupoid representation ShGT (XT) ≃ Sh(CT) then follows from descent theory. 2 Duality 2.1 Representation Theorem for Decidable Coherent Categories Since one can p... |

41 |
First Order Categorical Logic
- Makkai, Reyes
- 1977
(Show Context)
Citation Context ...nly a conservative extension of the theory and does not change the category of models, it is natural to represent the classical first-order theories by the subcategory of Boolean pretoposes (see e.g. =-=[15]-=-, [16]). We shall refer to the groupoids in the image of the semantic functor Mod restricted to the full subcategory of Boolean pretoposes BPTopκ � � �dCohκ, as Stone groupoids. Thus StoneGpd � � �Sem... |

22 |
Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories
- Lawvere
- 1963
(Show Context)
Citation Context ... algebraic theory T can be recovered (up to splitting of idempotents) from its categoryofmodelsintheformofthosefunctorsModT �Setswhichpreserve limits, filtered colimits, and regular epimorphisms (see =-=[2]-=-,[3]). Expanding from the algebraic case, recall, e.g. from [4, D1.1.], that the Horn formulas over a first-order signature are those formulas which are constructed using only connectives ⊤ and ∧. All... |

20 |
The classifying topos of a continuous groupoid
- Moerdijk
- 1988
(Show Context)
Citation Context ...an arbitrary topological groupoid, which we also write as H1 ⇉ H0, the topos of equivariant sheaves (or continuous actions) on H, written Sh(H) or ShH1(H0), consists of the following [4, B3.4.14(b)], =-=[11]-=-, [12]. An object of Sh(H) is a pair 〈a : A → H0,α〉, where a is a local homeomorphism(thatis, anobjectofSh(H0))andα : H1×H0A → Aisacontinuous function from the pullback (in Top) of a along the source ... |

13 | On the duality between varieties and algebraic theories, Algebra Universalis 49
- Adámek, Lawvere, et al.
- 2003
(Show Context)
Citation Context ...ebraic theory T can be recovered (up to splitting of idempotents) from its categoryofmodelsintheformofthosefunctorsModT �Setswhichpreserve limits, filtered colimits, and regular epimorphisms (see [2],=-=[3]-=-). Expanding from the algebraic case, recall, e.g. from [4, D1.1.], that the Horn formulas over a first-order signature are those formulas which are constructed using only connectives ⊤ and ∧. Allowin... |

11 |
Logical and cohomological aspects of the space of points of a topos
- Butz
- 1996
(Show Context)
Citation Context ... is the so-called classifying topos of (the Morleyization of) the theory, from which it is known that the theory can be recovered up to a notion of equivalence. (Here we build upon earlier results in =-=[1]-=- to the effect that any such topos can be represented by a topological groupoid constructed from its points. Our construction differs from the one given there in choosing a simpler cover which is bett... |

9 |
A theorem on Barr-exact categories, with an infinitary generalization
- Makkai
- 1990
(Show Context)
Citation Context ...ed existential quantification and pass to regular logic, then categories of models need no longer have arbitrary limits. But they still have products and filtered colimits, and, as shown by M. Makkai =-=[6]-=-, the definable set functors for regular formulas can now be characterized as those functors ModT → Sets that preserve precisely that. Adding the connectives ⊥ and ∨ to regular logic gives us the frag... |

7 |
duality for first order logic
- Stone
- 1987
(Show Context)
Citation Context ...they do have are ultra-products. Although ultra-products are not an intrin6sic feature of categories of models (for coherent theories), in the sense that they are not a categorical invariant, Makkai =-=[7]-=- shows that model categories and the category of sets can be equipped with a notion of ultra-product structure—turning them into so-called ultra-categories—which allows for the characterization of def... |

3 | First-order logical duality
- Awodey, Forssell
(Show Context)
Citation Context .... Finally, our construction restricts to classical Stone duality in the propositional case. Many more details of the results contained herein can be found in the second author’s doctoral dissertation =-=[9]-=-. 1 The Representation Theorem 1.1 Theories and Models We show how to recover a classical, first-order theory from its groupoid of models and model-isomorphisms, bounded in size and equipped with topo... |

2 |
Strong conceptual completeness for first-order logic
- Makkai
- 1988
(Show Context)
Citation Context ...conservative extension of the theory and does not change the category of models, it is natural to represent the classical first-order theories by the subcategory of Boolean pretoposes (see e.g. [15], =-=[16]-=-). We shall refer to the groupoids in the image of the semantic functor Mod restricted to the full subcategory of Boolean pretoposes BPTopκ � � �dCohκ, as Stone groupoids. Thus StoneGpd � � �SemGpd is... |

1 |
Sketches of an Elephant, vol. 43 and 44 of Oxford Logic Guides
- Johnstone
- 2002
(Show Context)
Citation Context ...coherent theory, TC, of C by having a sort for each object and a function symbol for eacharrow, andtakingasaxiomsallsequentswhicharetrueunderthecanonical interpretation ofthis languageinC (again, see =-=[4]-=- fordetails). Acoherent decidable category allows for the construction of a coherent decidable theory (including an inequality predicate for each sort), and Boolean coherent C allows for the construct... |

1 |
Duality and Definability in First Order Logic. No
- Makkai
- 1993
(Show Context)
Citation Context ...this approach can be modified in the case of classical first-order theories so that only the ultra-groupoids of models and isomorphisms, equipped with ultra-product structure, need be considered, see =-=[8]-=-. Our approach, similarly, relies on equipping the models of a theory with external structure, but in our case the structure is topological. We, too, restrict consideration to groupoids of models and ... |

1 |
Representing topoi by topologicalgroupoids
- Moerdijk
- 1998
(Show Context)
Citation Context ...esented as the topos of equivariant sheaves on a topological groupoid of ‘points’, or set-valued coherent functors, and invertible natural transformations. This builds upon earlier results in [1] and =-=[10]-=- to the effect that a coherent topos can be represented by a topological groupoid constructed from its points (our construction differs from the one given in loc.cit. in choosing a simpler cover which... |