Sets in HoTT Egbert Rijke Bas Spitters Radboud University Nijmegen - PowerPoint PPT Presentation

Introduction Regularity Quotients The object classifier Sets in HoTT Egbert Rijke Bas Spitters Radboud University Nijmegen April 23rd, 2013 Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Challenges of current type theories For finitary mathematics (Coq) type theory works very well. The extension to infinitary mathematics is challenging. No: quotients, functional extensionality, subset types... Univalence Axiom to the rescue! Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Setting ◮ Univalent foundations of mathematics. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Setting ◮ Univalent foundations of mathematics. ◮ We work with a univalent universe U. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Setting ◮ Univalent foundations of mathematics. ◮ We work with a univalent universe U. ◮ Types in U have the structure of weak ∞ -groupoids. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Setting ◮ Univalent foundations of mathematics. ◮ We work with a univalent universe U. ◮ Types in U have the structure of weak ∞ -groupoids. ◮ Sets (Set) in U are types for which UIP is valid. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Setting ◮ Univalent foundations of mathematics. ◮ We work with a univalent universe U. ◮ Types in U have the structure of weak ∞ -groupoids. ◮ Sets (Set) in U are types for which UIP is valid. ◮ Mere propositions (Prop) in U are types with proof irrelevance. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Do we have Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Do we have ◮ (Stable) image factorization, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Do we have ◮ (Stable) image factorization, ◮ quotients, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Do we have ◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Do we have ◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Do we have ◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, ◮ the collection axiom? Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Do we have ◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, ◮ the collection axiom? ◮ We will use ideas from Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Do we have ◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, ◮ the collection axiom? ◮ We will use ideas from ◮ Algebraic Set Theory (Joyal, Moerdijk). Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Aim of the current work ◮ To understand the category Set in a univalent universe. ◮ Initial question: Is Set a predicative topos? Do we have ◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, ◮ the collection axiom? ◮ We will use ideas from ◮ Algebraic Set Theory (Joyal, Moerdijk). ◮ Higher category theory (Lurie, Rezk) Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Squash ◮ The inclusion of Prop in U has a left adjoint called ( − 1)-truncation. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Squash ◮ The inclusion of Prop in U has a left adjoint called ( − 1)-truncation. � − � − 1 : U → Prop . Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Squash ◮ The inclusion of Prop in U has a left adjoint called ( − 1)-truncation. � − � − 1 : U → Prop . With unit | − | − 1 : A → � A � for every A : U. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Squash ◮ The inclusion of Prop in U has a left adjoint called ( − 1)-truncation. � − � − 1 : U → Prop . With unit | − | − 1 : A → � A � for every A : U. ◮ The universal property of ( − 1)-truncation is that ◮ For every type A : U, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Squash ◮ The inclusion of Prop in U has a left adjoint called ( − 1)-truncation. � − � − 1 : U → Prop . With unit | − | − 1 : A → � A � for every A : U. ◮ The universal property of ( − 1)-truncation is that ◮ For every type A : U, ◮ For every family P : � A � − 1 → Prop of mere propositions over � A � − 1 , Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier Squash ◮ The inclusion of Prop in U has a left adjoint called ( − 1)-truncation. � − � − 1 : U → Prop . With unit | − | − 1 : A → � A � for every A : U. ◮ The universal property of ( − 1)-truncation is that ◮ For every type A : U, ◮ For every family P : � A � − 1 → Prop of mere propositions over � A � − 1 , ◮ The pre-composition function � � λ s . λ a . s ( | a | − 1 ) : P ( x ) → P ( | a | − 1 ) a : A x : � A � − 1 is an equivalence. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier 0-truncation ◮ The inclusion of Set in U has a left adjoint called 0-truncation. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier 0-truncation ◮ The inclusion of Set in U has a left adjoint called 0-truncation. � − � 0 : U → Set . Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier 0-truncation ◮ The inclusion of Set in U has a left adjoint called 0-truncation. � − � 0 : U → Set . With unit | − | 0 : A → � A � 0 for every A : U. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier 0-truncation ◮ The inclusion of Set in U has a left adjoint called 0-truncation. � − � 0 : U → Set . With unit | − | 0 : A → � A � 0 for every A : U. ◮ The universal property of 0-truncation is that ◮ For every type A : U, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier 0-truncation ◮ The inclusion of Set in U has a left adjoint called 0-truncation. � − � 0 : U → Set . With unit | − | 0 : A → � A � 0 for every A : U. ◮ The universal property of 0-truncation is that ◮ For every type A : U, ◮ For every family P : � A � 0 → Set of sets over � A � 0 , Egbert Rijke, Bas Spitters Sets in HoTT