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

2018 ◽  
Vol 29 (4) ◽  
pp. 588-614 ◽  
Author(s):  
DAN FRUMIN ◽  
BENNO VAN DEN BERG
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Full Subcategory ◽  
Category Structure ◽  
Homotopy Category ◽  
Theoretic Model ◽  
Model Structure ◽  
Homotopy Equivalence ◽  
Homotopy Type Theory ◽  
Quillen Model

We present a way of constructing a Quillen model structure on a full subcategory of an elementary topos, starting with an interval object with connections and a certain dominance. The advantage of this method is that it does not require the underlying topos to be cocomplete. The resulting model category structure gives rise to a model of homotopy type theory with identity types, Σ- and Π-types, and functional extensionality. We apply the method to the effective topos with the interval object ∇2. In the resulting model structure we identify uniform inhabited objects as contractible objects, and show that discrete objects are fibrant. Moreover, we show that the unit of the discrete reflection is a homotopy equivalence and the homotopy category of fibrant assemblies is equivalent to the category of modest sets. We compare our work with the path object category construction on the effective topos by Jaap van Oosten.

Model Structures for the 𝔸1-homotopy Category

2019 ◽  
pp. 175-212
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Full Subcategory ◽  
Homotopy Category ◽  
Symmetric Power ◽  
Model Structure ◽  
Model Categories ◽  
Simplicial Presheaves ◽  
Basic Ideas ◽  
Quillen Model ◽  
Model Structures ◽  
Bousfield Localization

This chapter provides the 𝔸1-local projective model structure on the categories of simplicial presheaves and simplicial presheaves with transfers. These model categories, written as Δ‎opPshv(Sm)𝔸1 and Δ‎op PST(Sm)𝔸1, are first defined. Their respective homotopy categories are Ho(Sm) and the full subcategory DM eff nis ≤0 of DM eff nis. Afterward, this chapter introduces the notions of radditive presheaves and ̅Δ‎-closed classes, and develops their basic properties. The theory of ̅Δ‎-closed classes is needed because the extension of symmetric power functors to simplicial radditive presheaves is not a left adjoint. This chapter uses many of the basic ideas of Quillen model categories, which is a category equipped with three classes of morphisms satisfying five axioms. In addition, much of the material in this chapter is based upon the technique of Bousfield localization.

scholarly journals Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

scholarly journals Reviving the Philosophy of Geometry

2018 ◽  
Author(s):  
David Corfield
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Great Part ◽  
Mathematical Physics ◽  
Relevant Today ◽  
Homotopy Type Theory ◽  
Philosophical Account ◽  
Hard Times ◽  
Anglophone World

In the Anglophone world, the philosophical treatment of geometry has fallen on hard times. This chapter argues that philosophy can come to a better understanding of mathematics by providing an account of modern geometry, including its development of new forms of space, both for mathematical physics and for arithmetic. It returns to the discussions of Weyl and Cassirer on geometry whose concerns are very much relevant today. A way of encompassing a great part of modern geometry via homotopy toposes is discussed, along with the `cohesive’ variant of their internal language, known as `homotopy type theory’. With these tools in place, we can now start to see what an adequate philosophical account of current geometry might look like.

Homotopy Types

2020 ◽  
pp. 77-106
Author(s):  
David Corfield
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Definite Descriptions ◽  
Homotopy Types ◽  
Homotopy Type Theory

A further innovation is the introduction of an intensional type theory. Here one need not reduce equivalence to mere identity. How two entities are the same tells us more than whether they are the same. This is explained by the homotopical aspect of HoTT, where types are taken to resemble spaces of points, paths, paths between paths, and so on. This allows us to rethink Russell’s definite descriptions. Mathematicians use a ‘generalized the’ in situations where it appears that they might be talking about a multiplicity of instances, but there is essentially a unique instance. Taken together with the ‘univalence axiom’ there results a language in which anything that can be said of a type can be said of an equivalent type. This allows homotopy type theory to become the language of choice for a structuralist, avoiding the need for any kind of abstraction away from multiple instantiations.

scholarly journals Identity in Homotopy Type Theory, Part I: The Justification of Path Induction

2015 ◽  
Vol 23 (3) ◽  
pp. 386-406 ◽  
Author(s):  
James Ladyman ◽  
Stuart Presnell
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Homotopy Type Theory

scholarly journals Homotopy limits in type theory

2015 ◽  
Vol 25 (5) ◽  
pp. 1040-1070 ◽  
Author(s):  
JEREMY AVIGAD ◽  
KRZYSZTOF KAPULKIN ◽  
PETER LEFANU LUMSDAINE
Keyword(s):  
Homotopy Type ◽  
Systematic Study ◽  
Type Theory ◽  
Proof Assistant ◽  
Classical Approach ◽  
Homotopy Type Theory ◽  
The Coq Proof Assistant ◽  
Homotopy Limits ◽  
Coq Proof Assistant

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.

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