Periodic Reporting for period 1 - ForCUTT (Formalisation of Constructive Univalent Type Theory)
Reporting period: 2022-11-01 to 2025-04-30
There has been in the past 15 years remarkable achievements in the field of interactive theorem proving, both for checking complex software and non trivial mathematical proofs.
For software correctness, X. Leroy, (INRIA and College de France), has been leading since 2006 the CompCert project, developing a fully verified C compiler.
For mathematical proofs, these systems could handle highly complex arguments, such as the proof of the 4 colour theorem or the formal proof of Feit-Thompson Theorem.
More recently, the Xena project, lead by K. Buzzard, is developing a large library of mathematical facts, and has been able to help the mathematician P. Scholze (field medalist 2018) to check a highly non trivial proof.
All these examples have been carried out in systems based on the formalism of dependent type theory, and on early work of the PI. In parallel to these works, also around 15 years ago,
a remarkable and unexpected correspondance was discovered between this formalism and the abstract study of homotopy theory and higher categorical structures.
A special year 2012-2013 at the Institute of Advance Study (Princeton) was organised by the late V. Voevodsky (field medalist 2002, Princeton), S. Awodey (CMU) and the PI.
Preliminary results indicate that this research direction is productive, both for the understanding of dependent type systems and higher category theory, and suggest several crucial
open questions. The objective of this proposal is to analyse these questions, with the ultimate goal of formulating a new way to look at mathematical objects and potentially a new foundation of mathematics.
This could in turn be crucial for the design of future proof systems able to handle complex highly modular software systems and mathematical proofs.
For software correctness, X. Leroy, (INRIA and College de France), has been leading since 2006 the CompCert project, developing a fully verified C compiler.
For mathematical proofs, these systems could handle highly complex arguments, such as the proof of the 4 colour theorem or the formal proof of Feit-Thompson Theorem.
More recently, the Xena project, lead by K. Buzzard, is developing a large library of mathematical facts, and has been able to help the mathematician P. Scholze (field medalist 2018) to check a highly non trivial proof.
All these examples have been carried out in systems based on the formalism of dependent type theory, and on early work of the PI. In parallel to these works, also around 15 years ago,
a remarkable and unexpected correspondance was discovered between this formalism and the abstract study of homotopy theory and higher categorical structures.
A special year 2012-2013 at the Institute of Advance Study (Princeton) was organised by the late V. Voevodsky (field medalist 2002, Princeton), S. Awodey (CMU) and the PI.
Preliminary results indicate that this research direction is productive, both for the understanding of dependent type systems and higher category theory, and suggest several crucial
open questions. The objective of this proposal is to analyse these questions, with the ultimate goal of formulating a new way to look at mathematical objects and potentially a new foundation of mathematics.
This could in turn be crucial for the design of future proof systems able to handle complex highly modular software systems and mathematical proofs.
The work that conducted during this first period can be divided in the following way.
The main emphasis has been to explore further the surprising connections between the language of
dependent type theory and the notion of higher topos.
1. Constructive model of Univalent Type Theory
In joint work with Steve Awodey (CMU) and Emily Riehl (Johns Hopkins), we have completed the description of the Quillen model structure associated with a model of type theory and demonstrated its classical equivalence to spaces. The paper has been submitted and is available on arXiv. This is an important step in connecting dependent type theory with higher topos.
A limitation of this work, from a constructive perspective, is that the equivalence to spaces is established using classical logic. However, a significant breakthrough has been made by Christian Sattler, who shows that it is possible to construct a model that retains the expected correspondence to spaces within a constructive meta-theory. This model is obtained as a left exact localization of the presheaf model based on finite non-empty posets.
A large part of future work in this project will be on exploring further the properties of this model.
2. Prototype Implementation
My student Jonas Hofer implemented a type theory corresponding to Sattler's model.
It uses the duality between finite posets and finitely presented distributive lattices for obtaining a system with
decidable type checking.
3. Synthetic Mathematics
We have formulated a system of 3 axioms on top of dependent type theory with univalence that seems
complete for proving internal results valid in the Zariski higher topos. In particular, we get
an internal definition of schemes as functor of points.
We also get a new proof of the fact that the Picard group of the projective space is the group of integers.
The idea behind the new definition of propositional truncation used in this work was leveraged by
Rafael Bocquet to provide a new proof of Voevodsky's conjecture on homotopy canonicity.
A summary of ongoing developments can be found on the following link https://felix-cherubini.de/(opens in new window).
The main emphasis has been to explore further the surprising connections between the language of
dependent type theory and the notion of higher topos.
1. Constructive model of Univalent Type Theory
In joint work with Steve Awodey (CMU) and Emily Riehl (Johns Hopkins), we have completed the description of the Quillen model structure associated with a model of type theory and demonstrated its classical equivalence to spaces. The paper has been submitted and is available on arXiv. This is an important step in connecting dependent type theory with higher topos.
A limitation of this work, from a constructive perspective, is that the equivalence to spaces is established using classical logic. However, a significant breakthrough has been made by Christian Sattler, who shows that it is possible to construct a model that retains the expected correspondence to spaces within a constructive meta-theory. This model is obtained as a left exact localization of the presheaf model based on finite non-empty posets.
A large part of future work in this project will be on exploring further the properties of this model.
2. Prototype Implementation
My student Jonas Hofer implemented a type theory corresponding to Sattler's model.
It uses the duality between finite posets and finitely presented distributive lattices for obtaining a system with
decidable type checking.
3. Synthetic Mathematics
We have formulated a system of 3 axioms on top of dependent type theory with univalence that seems
complete for proving internal results valid in the Zariski higher topos. In particular, we get
an internal definition of schemes as functor of points.
We also get a new proof of the fact that the Picard group of the projective space is the group of integers.
The idea behind the new definition of propositional truncation used in this work was leveraged by
Rafael Bocquet to provide a new proof of Voevodsky's conjecture on homotopy canonicity.
A summary of ongoing developments can be found on the following link https://felix-cherubini.de/(opens in new window).
We believe that the work on Synthetic Algebraic Geometry, presenting a simple system of 3 axioms for the Zariski higher topos, is an important achievement.