[Paper Review] A 2-Categorical Analysis of the Tripos-to-Topos Construction

Jonas Frey|arXiv (Cornell University)|Apr 14, 2011

Homotopy and Cohomology in Algebraic Topology23 references13 citations

TL;DR

This paper provides a 2-categorical characterization of the tripos-to-topos construction as a biadjunction in a bicategory of equipment-like structures, resolving functoriality issues by using oplax functors. The key contribution is a decomposition of the construction into two steps: first to a weakened quasitopos (q-topos), then to a topos, enabling a systematic framework for analyzing geometric morphisms and subtoposes via tripos morphisms that preserve only finite limits and meets.

ABSTRACT

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.

Motivation & Objective

• To provide a universal, abstract characterization of the tripos-to-topos construction that accommodates arbitrary geometric morphisms, not just regular ones.
• To resolve the issue of non-functoriality in the original construction by using oplax functors on certain classes of cartesian functors between triposes.
• To decompose the tripos-to-topos construction into two steps: first to a q-topos (a weakened quasitopos), then to a topos, enabling clearer structural analysis.
• To establish a framework where questions about functors between toposes induced by triposes can be reduced to questions about morphisms between the underlying triposes.
• To generalize existing approaches by using higher-dimensional category theory, particularly bicategories and equipments, to handle the logical and categorical structure of triposes and toposes.

Proposed method

• The paper employs a bicategory enriched category of equipment-like structures to model the 2-categorical framework necessary for handling oplax functors in the tripos-to-topos construction.
• It introduces a biadjunction F ⊣ S between the 2-category of triposes and the 2-category of q-toposes, capturing the first step of the construction.
• It establishes a second biadjunction T ⊣ U between q-toposes and toposes, completing the decomposition of the full tripos-to-topos process.
• The construction is analyzed using 2-dimensional categorical tools, including fibrational completions and the internal logic of triposes, to handle non-regular morphisms.
• It uses non-extensional higher-order intuitionistic logic (with explicit contexts and equality rules) to formalize the internal language of triposes and their morphisms.
• The framework allows for the systematic study of geometric morphisms and subtoposes via corresponding tripos morphisms that preserve only finite limits and finite meets.

Experimental results

Research questions

• RQ1 How can the tripos-to-topos construction be universally characterized in a way that accommodates all geometric morphisms, not just regular ones?
• RQ2 What is the precise 2-categorical structure underlying the tripos-to-topos construction, particularly regarding its functoriality?
• RQ3 Can the tripos-to-topos construction be decomposed into intermediate steps that clarify its categorical and logical components?
• RQ4 How do morphisms between triposes induce functors between the resulting toposes, especially when only finite limits and meets are preserved?
• RQ5 What role do q-toposes play as an intermediate structure between triposes and toposes in this decomposition?

Key findings

• The tripos-to-topos construction is characterized as a composite of two biadjunctions: F ⊣ S between triposes and q-toposes, and T ⊣ U between q-toposes and toposes.
• The intermediate structure, q-toposes, is identified as a weakened version of a quasitopos, providing a natural intermediate stage in the construction.
• The construction is only oplax functorial on certain classes of cartesian functors between triposes, necessitating the use of 2-categorical tools like biadjunctions.
• The framework successfully handles non-regular morphisms (preserving only finite limits and meets), resolving limitations of prior 2-categorical approaches.
• The analysis provides a systematic method to reduce questions about functors between toposes to questions about morphisms between underlying triposes.
• The paper demonstrates that geometric morphisms and subtoposes correspond to analogous constructs between triposes, enabling more tractable calculations at the tripos level.