A homotopy-theoretic model of function extensionality in the effective topos

Frumin, Daniil; Berg, Benno van den

doi:10.48550/arxiv.1701.08369

Cited by 2 publications

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, for this object we have that I → 1 is a trivial fibration and that P X ≃ X I , while J → 1 is far from a trivial fibration and we have X ≃ P X ≃ X J only if X is discrete. Since J is a version of ∇(2), this also shows that our work here is rather different in spirit from that in [17] and [9]. Another difference is that these papers seek to define homotopy-theoretic structures on the effective topos and do not regard it as a homotopy-theoretic quotient of some other category.…”

Section: Discrete Fibrations In Effmentioning

confidence: 93%

Univalent polymorphism

Berg

2018

Preprint

Self Cite

View full text Add to dashboard Cite

We show that Martin Hyland's effective topos can be exhibited as the homotopy category of a path category EFF. Path categories are categories of fibrant objects in the sense of Brown satisfying two additional properties and as such provide a context in which one can interpret many notions from homotopy theory and Homotopy Type Theory. Within the path category EFF one can identify a class of discrete fibrations which is closed under push forward along arbitrary fibrations (in other words, this class is polymorphic or closed under impredicative quantification) and satisfies propositional resizing. This class does not have a univalent representation, but if one restricts to those discrete fibrations whose fibres are propositions in the sense of Homotopy Type Theory, then it does. This means that, modulo the usual coherence problems, it can be seen as a model of the Calculus of Constructions with a univalent type of propositions. We will also build a more complicated path category in which the class of discrete fibrations whose fibres are sets in the sense of Homotopy Type Theory has a univalent representation, which means that this will be a model of the Calculus of Constructions with a univalent type of sets.

show abstract

Section: Discrete Fibrations In Effmentioning

confidence: 93%

Univalent polymorphism

Berg

2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, although the Hofmann-Streicher [HS99] universe construction (the basis for the construction in Section 8.2 of [CCHM18] of a fibrant universe satisfying the full univalence axiom) can be extended from presheaf to sheaf toposes via the use of sheafification [Str05, Section 3], it seems that sheafification does not interact well with the CCHM notion of fibration. In another direction, recent work of Frumin and Van Den Berg [Fv18] makes use of our elementary, axiomatic approach using a non-Grothendieck topos, namely the effective topos [Hyl82].…”

Section: Theorem 73 (Converting Equivalences To Paths)mentioning

confidence: 99%

Axioms for Modelling Cubical Type Theory in a Topos

Orton

Pitts

2018

Logical Methods in Computer Science ; Volume 14

View full text Add to dashboard Cite

The homotopical approach to intensional type theory views proofs of equality as paths. We explore what is required of an object I in a topos to give such a path-based model of type theory in which paths are just functions with domain I. Cohen, Coquand, Huber and Mörtberg give such a model using a particular category of presheaves. We investigate the extent to which their model construction can be expressed in the internal type theory of any topos and identify a collection of quite weak axioms for this purpose. This clarifies the definition and properties of the notion of uniform Kan filling that lies at the heart of their constructive interpretation of Voevodsky's univalence axiom.

show abstract

A homotopy-theoretic model of function extensionality in the effective topos

Cited by 2 publications

References 0 publications

Univalent polymorphism

Univalent polymorphism

Axioms for Modelling Cubical Type Theory in a Topos

Contact Info

Product

Resources

About