Remarks on the Tripos To Topos Construction: extensionality, comprehensions, quotients and cauchy-complete objects

通过Tripos到Topos构造对局部层化的代数抽象

An Algebraic Abstraction of the Localic Sheafification via the Tripos-to-Topos Construction

本文通过代数抽象局部预层与层化，并借助比较引理联系locale超紧化，为局部拓扑斯与可实现性拓扑斯建立统一的几何框架。

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

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 $\nabla 2$. In the resulting model structure we identify uniform inhabited objects as contractible …

A 2-Categorical Analysis of the Tripos-to-Topos Construction

We characterize the tripos-to-topos construction of Hyland, Johnstone and Pitts as a biadjunction in a bicategory enriched category of equipment-like structures. These abstract concepts are necessary to handle the presence of oplax constructs --- the construction is only oplax functorial on certain classes of cartesian functors between triposes. A by-product of our analysis is the decomposition of the tripos-to-topos construction into two steps, the intermediate step being a weakened version of quasitoposes.

A tripos based on compact Hausdorff spaces

The category of compact Hausdorff spaces is the base of tripos. As such it can be freely completed to an elementary topos.

Implicative models of intuitionistic set theory

In this paper we will show that using implicative algebras one can produce models of intuitionistic set theory generalizing both realizability and Heyting-valued models. This has as consequence that if one assumes the inaccessible cardinal axiom, then every topos which is obtained from a Set-based tripos as the result of the tripos-to-topos construction hosts a model of intuitionistic set theory.

一类用于修正可实现性的箭头代数

A category of arrow algebras for modified realizability

该研究定义箭头代数的态射概念以对应关联tripos的态射，用底层代数的核刻画箭头tripos的子tripos，还将修正可实现性拓展至箭头代数并证明其函子性，推进了相关代数与topos理论研究。