Papers by Wouter W P Stekelenburg
This paper introduces categories of assemblies which are closely connected to realizability inter... more This paper introduces categories of assemblies which are closely connected to realizability interpretations and which are based on an important subcategory of the effective topos. There is a list of properties which characterize these categories of assemblies up to equivalence.
This paper shows constructive alternatives for a couple of theorems that make simplicial sets a m... more This paper shows constructive alternatives for a couple of theorems that make simplicial sets a model of homotopy type theory. The constructive theories are valid in the exact completion of the category of assemblies. Ultimately this shows that simplicial PERs are a model of homotopy type theory.
In the paper "Extensional PERs" by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a category $\mat... 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 endofunctors of the category. We extend that result to include all internal functors on $\mathcal{C}$ when $\mathcal{C}$ is viewed as a full internal category of the effective topos. This is done using two general results: one about internal functors in general, and one about internal functors in the effective topos.
Relative realizability toposes satisfy a universal property that involves regular functors to oth... more Relative realizability toposes satisfy a universal property that involves regular functors to other categories. We use this universal property to define what relative realizability categories are, when based on other categories than of the topos of sets. This paper explains the property and gives a construction for relative realizability categories that works for arbitrary base Heyting categories. The universal property shows us some new geometric morphisms to relative realizability toposes too.
Many ex/reg completions J : C → C ex/reg that arise in categorical realizability and tripos theor... more Many ex/reg completions J : C → C ex/reg that arise in categorical realizability and tripos theory admit left Kan extensions of arbitrary finitely continuous functors to arbitrary exact categories. This paper identifies the property which is responsible for these extensions: the functors are resolvent. Resolvency is characteristic of toposes that are ex/reg completions of regular categories with (weak) dependent products and generic monomorphisms. It also helps to characterize the toposes that the tripos-to-topos construction produces.
In the paper "Extensional PERs" by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a category C of ... more In the paper "Extensional PERs" by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a category C of "pointed complete extensional PERs" and computable maps is introduced to provide an instance of an 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 endofunctors of the category. We extend that result to include all internal functors on C when C is viewed as a full internal category of the effective topos. This is done using two general results: one about internal functors in general, and one about internal functors in the effective topos.
The relative realizability toposes that Awodey, Birkedal and Scott introduced in [1] satisfy a un... more The relative realizability toposes that Awodey, Birkedal and Scott introduced in [1] satisfy a universal property that involves regular functors to other categories. We use this universal property to define what relative realizability categories are, when based on other categories than of the topos of sets. This paper explains the property and gives a construction for relative realizability categories that works for arbitrary base Heyting categories. The universal property shows us some new geometric morphisms to relative realizability toposes too.
Drafts by Wouter W P Stekelenburg
This paper aims to help the development of new models of homotopy type theory, in particular with... more This paper aims to help the development of new models of homotopy type theory, in particular with models that are based on realizability toposes. For this purpose it develops the foundations of an internal simplicial homotopy that does not rely on classical principles that are not valid in realizability toposes and related categories.
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Papers by Wouter W P Stekelenburg
Drafts by Wouter W P Stekelenburg