## Model-theoretic imaginaries and coherent sheaves (2006)

Citations: | 4 - 3 self |

### BibTeX

@MISC{Rajani06model-theoreticimaginaries,

author = {Ravi Rajani and Mike Prest},

title = {Model-theoretic imaginaries and coherent sheaves},

year = {2006}

}

### OpenURL

### Abstract

### Citations

503 | Categories for the Working Mathematician - MacLane - 1971 |

219 | Locally presentable and accessible categories - Adámek, Rosický - 1994 |

71 | Model Theory and Modules - Prest - 1988 |

69 | I.: Preprojective modules over artin algebras - Auslander, Reiten - 1980 |

49 |
Model theory. Third edition
- Chang, Keisler
- 1990
(Show Context)
Citation Context ...∀x ⎛ ⎝ � � gij(x) → ∃y � ⎞⎫ � ⎬ hij(x, y) ⎠ i∈I ′ j∈J i∈I ′ j∈J ⎭ I ′ ⊆I Now a definable subcategory D is an elementary class closed under directed colimits. The fact that D is elementary follows (by =-=[9]-=-, Corollary 6.1.16) from the fact that D is closed under ultraproducts and its complement is closed under ultrapowers (since every structure is purely, even elementarily, embeddable in any of its ultr... |

44 |
First order categorical logic
- Makkai, Reyes
- 1977
(Show Context)
Citation Context ...theorem. Taking model theory rather than topos theory as our starting point, we set out to prove a non-abelian version of Burke’s result for modules. Being aware of, but not conversant with, the book =-=[13]-=- of Makkai and Reyes, it was only later that we realised the relevance of their work. Fix an arbitrary (finitary) first-order language L and let C denote the category of L-structures. Recall that a co... |

20 |
Elementary duality of modules
- Herzog
- 1993
(Show Context)
Citation Context ...e category (Mod-R) eq+ of pp-defined imaginaries for R-modules. Imaginaries belong to set-based model theory. Indeed, thinking of them as forming a category was a novel step which was taken by Herzog =-=[6]-=- when he defined the category (Mod-R) eq+ . This equivalence has the consequence that there are two quite different languages for talking about the same circle of ideas. More important, it allows an e... |

19 | The first order properties of algebraic systems - Feferman, Vaught - 1959 |

14 | Some Model-Theoretic Properties of Functor Categories for Modules - Burke - 1994 |

13 | Functors on Locally Finitely Presented Categories - Krause - 1998 |

12 | Spectra and Localisation - Prest, Purity |

6 |
Model Theory, Encyclopedia of
- Hodges
- 1993
(Show Context)
Citation Context ...limits. � The above proof shows, amongst other things, that the interpretation functor Γ takes positive existential formulas to their syntactic translations in the sense of the Reduction Theorem (see =-=[7]-=-). We will henceforth refer to the following restricted form of this theorem as the Reduction Theorem. Its proof is a direct consequence of the above proof and the proof in [7]. Theorem 2 (Reduction T... |

5 |
Theory of models with generalized atomic formulas
- Keisler
- 1960
(Show Context)
Citation Context ...particular sort. Theorem 3 The following are equivalent. (i) D is a definable subcategory; (ii) D can be axiomatised by a coherent theory. This result is a direct consequence of a result of Keisler’s =-=[8]-=-. Recall that a coherent theory is one whose axioms are of the form ∀x(ϕ(x) → ψ(x)) where ϕ and ψ are positive existential. Keisler’s result is that an elementary class of structures is closed under d... |

2 | Handbook of Categorical Algebra, Vols 1-3 - Borceux - 1994 |