The Weak Choice Principle WISC may Fail in the Category of Sets

Roberts, David

doi:10.1007/s11225-015-9603-6

Cited by 8 publications

(7 citation statements)

References 8 publications

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“…Such toposes are models for set theory without the Axiom of Choice, and categories internal to them are simply small categories. Examples are given by the categories of sets in models of ZF as given by Gitik (cf [15]) and Karagila [4], or the topos constructed in the author's [10].…”

Section: Resultsmentioning

confidence: 99%

On Certain 2-Categories Admitting Localisation by Bicategories of Fractions

Roberts¹

2015

Appl Categor Struct

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Pronk's theorem on bicategories of fractions is applied, in almost all cases in the literature, to 2-categories of geometrically presentable stacks on a 1-site. We give an proof that subsumes all previous such results and which is purely 2-categorical in nature, ignoring the nature of the objects involved. The proof holds for 2-categories that are not (2,1)-categories, and we give conditions for local essential smallness. This is the published version of arXiv:1402.7108.

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Section: Resultsmentioning

confidence: 99%

On Certain 2-Categories Admitting Localisation by Bicategories of Fractions

Roberts¹

2015

Appl Categor Struct

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“…Thus starting from the category of sets in classical set theory with AC, it holds in the kinds of topos that have been used to model type theory with various kinds of higher inductive types, whose semantics motivates the work presented here. (However, WISC does not hold in all toposes, as Roberts [27] shows.) For the theorem below we need to use the double cover signature ∆ X of a set X ∈ Set, inspired by the use that Swan [31] makes of WISC; see also [14,Definition 5.6].…”

Section: Initial Algebras For Sized Endofunctorsmentioning

confidence: 98%

Constructing Initial Algebras Using Inflationary Iteration

Pitts

Steenkamp

2021

Preprint

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An old theorem of Adámek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using "inflationary" iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor's constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich collection of endofunctors to which the new theorem applies, provided one admits a weak form of choice (the WISC axiom of van den Berg, Moerdijk, Palmgren and Streicher) that is known to hold, for example, in the internal logic of very many kinds of elementary topos.

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“…Note that in classical set theory AC implies WISC: the axiom of choice implies that covers split and so we can take Cov A to be the type 1 with unique element 0 ∶ 1 and Dom A 0 = A. WISC is independent of ZF in classical logic [Kar14;Rob15]. In constructive logic it is preserved by many ways of making new toposes from existing ones, in particular by the formation of sheaf toposes and realizability toposes over a given topos [vdBM14].…”

Section: Weakly Initial Sets Of Coversmentioning

confidence: 99%

Quotients, inductive types, and quotient inductive types

Fiore¹,

Pitts²,

Steenkamp³

2021

Preprint

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This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly infinitary operators and equations. We prove that QWI types can be derived from quotient types and inductive types in the type theory of toposes with natural number object and universes, provided those universes satisfy the Weakly Initial Set of Covers (WISC) axiom. We do so by constructing QWI types as colimits of a family of approximations to them defined by well-founded recursion over a suitable notion of size, whose definition involves the WISC axiom. We developed the proof and checked it using the Agda theorem prover.

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The Weak Choice Principle WISC may Fail in the Category of Sets

Cited by 8 publications

References 8 publications

On Certain 2-Categories Admitting Localisation by Bicategories of Fractions

On Certain 2-Categories Admitting Localisation by Bicategories of Fractions

Constructing Initial Algebras Using Inflationary Iteration

Quotients, inductive types, and quotient inductive types

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