A modest Kan complex is a modest simplicial set which has a right lifting property with respect t... more A modest Kan complex is a modest simplicial set which has a right lifting property with respect to horn inclusions $\Lambda_k[n] \to \Delta[n]$. This paper develops the categorical logical that is required to show that there is a univalent universe of modest Kan complexes among simplicial assemblies.
This is the author's Ph.D. Thesis. It contains results from four years of research into reali... more This is the author's Ph.D. Thesis. It contains results from four years of research into realizability and categorical logic. The main subjects are the axiomatisation of realizable propositions, and a characterization of realizability categories as pseudoinitial objects. Realizability is a collection of techniques in the study of constructive logic. Some forms of realizability induce realizability categories, which are Heyting categories and therefore have a first order intuitionistic logic as internal language. The axiomatisation chapter of the thesis explains how and to what extend we can axiomatise the set of valid propositions in this internal language. The realizability categories chapter explains how to find regular functors from realizability categories into other categories.
In the paper "Extensional PERs" by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a cate... more In the paper "Extensional PERs" by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a category $\mathcal{C}$ of "pointed complete extensional PERs" and computable maps is introduced to provide an instance of an \emph{algebraically compact category} relative to a restricted class of functors. Algebraic compactness is a synthetic condition on a category which ensures solutions of recursive equations involving
The relative realisability toposes introduced by Awodey, Birkedal and Scott in Awodey et al. (200... more The relative realisability toposes introduced by Awodey, Birkedal and Scott in Awodey et al. (2002) satisfy a universal property involving regular functors to other categories. We use this universal property to define what relative realisability categories are when they are based on categories other than the topos of sets. This paper explains the property and gives a construction for relative realisability categories that works for arbitrary base Heyting categories. The universal property also provides some new geometric morphisms to relative realisability toposes.
A modest Kan complex is a modest simplicial set which has a right lifting property with respect t... more A modest Kan complex is a modest simplicial set which has a right lifting property with respect to horn inclusions $\Lambda_k[n] \to \Delta[n]$. This paper shows that there is a univalent universe of modest Kan complexes among simplicial assemblies.
A modest Kan complex is a modest simplicial set which has a right lifting property with respect t... more A modest Kan complex is a modest simplicial set which has a right lifting property with respect to horn inclusions $\Lambda_k[n] \to \Delta[n]$. This paper develops the categorical logical that is required to show that there is a univalent universe of modest Kan complexes among simplicial assemblies.
This is the author's Ph.D. Thesis. It contains results from four years of research into reali... more This is the author's Ph.D. Thesis. It contains results from four years of research into realizability and categorical logic. The main subjects are the axiomatisation of realizable propositions, and a characterization of realizability categories as pseudoinitial objects. Realizability is a collection of techniques in the study of constructive logic. Some forms of realizability induce realizability categories, which are Heyting categories and therefore have a first order intuitionistic logic as internal language. The axiomatisation chapter of the thesis explains how and to what extend we can axiomatise the set of valid propositions in this internal language. The realizability categories chapter explains how to find regular functors from realizability categories into other categories.
In the paper "Extensional PERs" by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a cate... more In the paper "Extensional PERs" by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a category $\mathcal{C}$ of "pointed complete extensional PERs" and computable maps is introduced to provide an instance of an \emph{algebraically compact category} relative to a restricted class of functors. Algebraic compactness is a synthetic condition on a category which ensures solutions of recursive equations involving
The relative realisability toposes introduced by Awodey, Birkedal and Scott in Awodey et al. (200... more The relative realisability toposes introduced by Awodey, Birkedal and Scott in Awodey et al. (2002) satisfy a universal property involving regular functors to other categories. We use this universal property to define what relative realisability categories are when they are based on categories other than the topos of sets. This paper explains the property and gives a construction for relative realisability categories that works for arbitrary base Heyting categories. The universal property also provides some new geometric morphisms to relative realisability toposes.
A modest Kan complex is a modest simplicial set which has a right lifting property with respect t... more A modest Kan complex is a modest simplicial set which has a right lifting property with respect to horn inclusions $\Lambda_k[n] \to \Delta[n]$. This paper shows that there is a univalent universe of modest Kan complexes among simplicial assemblies.
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