M. Menni. Exact Completions and Toposes. PhD thesis, University of Edinburgh, 2000. Home/Search Document Details and Download
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Inductive Types and Exact Completion - van den Berg (2002) (Correct)
....familiar from the theory of exact completions, will repeatedly be used. In the exact completion, the objects in the image of y are, up to iso, the projectives of this category. And every object in the exact completion can be covered with a regular epi by such a projective. See, for instance, [5]. Lemma 4.6 The canonical map q in C ex =A is epi. The fact that q 1 is canonical means that the diagram jW R Psi Gamma Gamma commutes. Proof : Since w was a weak P f algebra, we know that W was a weak version of in C=A. We can now define the equivalence ....
M. Menni. Exact completions and toposes. Ph.D. thesis, University of Edinburgh, 2000.
Realizability: A Historical Essay - van Oosten (2000) (1 citation) (Correct)
....explanation of when (if) C) ex=lex is a topos, see [67] 32 The constructions ex lex and ex reg are well explained in [17] and [12] 33 [67] has an independent, abstract argument that Asm ex=lex is a topos. He obtains a whole hierarchy of toposes, starting with Eff and Asm ex=lex . See also [68] 18 is Eff A in A Longley defines a 2 category Pca of partial combinatory algebras, such that the category Pca(A; B) is equivalent to the category of exact functors Eff A Eff B which commute with the inclusions from Sets into these toposes. At first sight, his definition looks like a hack, ....
.... category of T 0 topological spaces; in fact, it is the regular completion of T 0 spaces ( 83] The relationships between these various completions, and when they have nice properties (being locally cartesian closed or toposes) have been systematically studied by Mat ias Menni in his thesis ([68]) obtaining a synthesis of all previous work in this area. 2.4 Axiomatization Revisited In his seminal paper [40] Hyland had finished with the comment: What we lack, above all [ is any real information analogous to the results obtained in Troelstra ( 93] axiomatizing realizability ....
M. Menni. Exact Completions and Toposes. PhD thesis, University of Edinburgh, 2000.
Realizability Toposes and Ordered PCA's - Hofstra, van Oosten (2001) (Correct)
....basic ingredient for the construction of various realizability toposes, of which the Effective Topos is undoubtedly the most famous. There is more than one way to present the realizability topos associated to a pca; one may take the exact completion of the category of partitioned assemblies (see [7]) or one can use tripos theory. Triposes built from pca s are, together with those from locales, the most important and most extensively studied instances of triposes, but from a structural point of view, there are important differences between the two; whereas locales are organized in a ....
....section is devoted to a brief study of the occurrence of local localic maps between realizability toposes. This is essentially a translation of the work done in [1] into our own framework. Finally, section 4 deals with an application to chains of inclusions of realizability toposes. In his thesis [7], Menni discusses hierarchies of toposes of the form (C reg(n) ex , where C is a category satisfying some conditions, Gamma) reg(n) denotes the n fold regular completion, and ( Gamma) ex denotes the exact completion. He conjectured, that there should be a tripos theoretic presentation of this ....
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M. Menni. Exact Completions and Toposes. PhD thesis, University of Edinburgh, 2000.
Ordered Partial Combinatory Algebras - Pieter Hofstra And (Correct)
.... topos RT[B ; A ] coincides with RT[B ] A second application is a presentation of a hierarchy of realizability toposes, induced by the sequence of opca s A ; T A ; T 2 A ; The fact that certain hierarchies can be presented in this tripos theoretic way was already conjectured by Menni [9]. Finally, we give a slight generalization of a theorem by Johnstone and Robinson, stating that the Effective Topos is not equivalent to any topos obtained from a total combinatory algebra. 2 Definitions and Basic Properties This section sets out the definitions and reviews basic properties. We ....
....ffl Z 0 is of the required form. Before we state the next theorem, we recall that a diagram of the form (X; ffl X ) f jX (Y; ffl Y ) jY r(X) r(f) r(Y ) is a pullback if and only if (X; ffl X ) X; ffl 0 ) with ffl 0 (x) ffl Y (f(x) for all x 2 X. Following Menni (see [9]) we call such maps pre embeddings. 13 Theorem 4.4 For any opca A , the following are equivalent: 1. PAss(A ) is a regular completion; 2. There is a cylinder of adjoints a OE a ffi : PAss(A ) Ass(A ) with OE ffi ffi Id, and with preserving finite limits and commuting with the ....
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M. Menni. Exact Completions and Toposes. PhD thesis, University of Edinburgh, 2000.
A Characterization Of The Left Exact Categories Whose Exact.. - Menni (1999) (3 citations) Self-citation (Menni) (Correct)
....CATEGORIES WHOSE. 5 Y # # Y # # # # # # # # f # X # # f # # # f # # # # It is worth mentioning that every epi splits in the presence of a strong proof classifier (i.e. a weak proof classifier for which the # f is required to be unique) The details of this can be found in [20] where weak proof classifiers are called generic proofs. With this terminology we are ready to state our characterization. Theorem 1. C ex is a topos if and only if C has weak dependent products and a weak proof classifier. The rest of the section is devoted to the proof of this fact. First we ....
....is a topos. As we mentioned, the category of assemblies is the regular completion of PA . It is actually possible to iterate the production of toposes in this way in the sense that there exist conditions on C that ensure that C ex is a topos and that C reg also satisfies the same conditions (see [20]) 5.2. Presheaf toposes. It is well known that many presheaf toposes arise as exact completions. In this section we review this fact (which can be proved without our characterization) and give explicit constructions of the weak proof classifiers involved. In order to present presheaf toposes as ....
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M. Menni. Exact completions and toposes. PhD thesis, University of Edinburgh, 2000.
Relative Completions - Hofstra (2002) (Correct)
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M. Menni. Exact Completions and Toposes. PhD thesis, University of Edinburgh, 2000.
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