On a Generalization of Equilogical Spaces

Pasquali, Fabio

doi:10.1007/s11787-018-0192-9

Cited by 4 publications

(2 citation statements)

References 13 publications

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“…The change of base of triposes of the form P H along the forgetful functor Top G G Set is again a tripos as it suffices to endow H A with the indiscrete topology to have the power objects (see [Pas18]). In the special case of H = {0, 1}, the tripos P H reduces to the contravariant powerset functor P: Set op G G ISL and its change of base along Top G G Set is a tripos that we call T .…”

Section: Applicationsmentioning

confidence: 99%

Quasi-toposes as elementary quotient completions

Maietti¹,

Pasquali²,

Rosolini³

2021

Preprint

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The elementary quotient completion of an elementary doctrine in the sense of Lawvere was introduced in previous work by the first and third authors. It generalises the exact completion of a category with finite products and weak equalisers. In this paper we characterise when an elementary quotient completion is a quasi-topos. We obtain as a corollary a complete characterisation of when an elementary quotient completions is an elementary topos. As a byproduct we determine also when the elementary quotient completion of a tripos is equivalent to the doctrine obtained via the tripos-to-topos construction.Our results are reminiscent of other works regarding exact completions and put those under a common scheme: in particular, Carboni and Vitale's characterisation of exact completions in terms of their projective objects, Carboni and Rosolini's characterisation of locally cartesian closed exact completions, also in the revision by Emmenegger, and Menni's characterisation of the exact completions which are elementary toposes.The paper contains results presented by the authors at several international meetings in the past years, in particular at Logic Colloquium 2016 and Category Theory 2017, and during the Trimester devoted to Types, Homotopy Type Theory and Verification at the Hausdorff research Institute for Mathematics in 2018. We would like to thank the organisers of these events who gave us the opportunity to present our results.

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Section: Applicationsmentioning

confidence: 99%

Quasi-toposes as elementary quotient completions

Maietti¹,

Pasquali²,

Rosolini³

2021

Preprint

Self Cite

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“…Remark 4.15. The category Equ of equilogical spaces introduced in [Sco96] is the domain of the elementary quotient completion of the doctrine of subspace inclusions on Top 0 , see [MPR17], and also [Pas18a] where a more general situation is considered. In the same vein, one can show that Equ is the full and reflective subcategory of (Top 0 ) ex/lex on those objects whose equality predicate is stable under the double negation [Ros00].…”

Section: :14mentioning

confidence: 99%

Elementary Quotient Completions, Church's Thesis, and Partioned Assemblies

Maietti¹,

Pasquali²,

Rosolini³

2019

Logical Methods in Computer Science ; Volume 15

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Hyland's effective topos offers an important realizability model for constructive mathematics in the form of a category whose internal logic validates Church's Thesis. It also contains a boolean full sub-quasitopos of "assemblies" where only a restricted form of Church's Thesis survives. In the present paper we compare the effective topos and the quasitopos of assemblies each as the elementary quotient completions of a Lawvere doctrine based on the partitioned assemblies. In that way we can explain why the two forms of Church's Thesis each category satisfies differ by the way each is inherited from specific properties of the doctrine which determines the elementary quotient completion.Asm for the full subcategory they determine, see [CFS88,vO08]. In the category Asm the

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Universal Logic: Evolution of a Project

Béziau

2018

Log. Univers.

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On a Generalization of Equilogical Spaces

Cited by 4 publications

References 13 publications

Quasi-toposes as elementary quotient completions

Quasi-toposes as elementary quotient completions

Elementary Quotient Completions, Church's Thesis, and Partioned Assemblies

Universal Logic: Evolution of a Project

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