Computational Paths, identitytype, and the groupoid model

LSFA 2015 Arthur Freitas Ramos

Ruy de Queiroz

Anjolina de Oliveira

Motivation

• Mathematics becoming increasingly abstract and complex

• Automatic Proof Checkers

• Type Theory: Identity type as a bridge between computer science andmath

• Objective: To propose a more intuitive approach to the identity type

Intensional Identity Type

• Witness p as proof of propositional equality between two objects of thesame type

• Connection between extensionality and intensionality

• Homotopy Type Theory: Semantic connection with homotopy (paths)

• HOFMANN - STREICHER 1994: Groupoid structure refutes the principle ofthe unicity of proofs of equality

• LUMSDAINE 2009: weak structure

Identity Type: Formal Construction

Identity Type: New Approach

• We propose a new approach to the identity type

• Objective: to be more intuitive than the classic approach

• Based on Computational Paths, entity originally proposed by Gabbay and Ruy de Queiroz in 1994

• Paths as part of the syntax of type theory: algebra of paths (or calculus of paths)

• Main objectives of this work: Detailed explanation of our new approach andproof of essential properties, such as the groupoid structure

Beta Equality

• Given terms P e Q if we say that they are if:

• reduction:

• reduction together with : theory of

Theory of Lambda-Beta-Eta Equality

• Axiom:

• Inference rules:

Equality Theory for the Product Type

Computational Paths

• Composition of axioms and inference rules s that establishes thepropositional equality between two terms a : A and b : A

• Notation a =s b : A

• Composition done by applications of the transitivity

Computational Paths: Example

• Path between and :

From we obtain

From we obtain

To obtain the final path between and , we just need toconcatenate both paths using the transitivity, obtaining:

Type Identity as the Type of ComputationalPaths: Formalization

Type Identity as the Type of ComputationalPaths: Formalization

Usage example: Symmetry

• Construction of :

Usage example: Transitivity

• Construction of :

Term Rewriting System – LNDEQ-TRS

• Reduction between different paths

• Simple examples: and ; and

• Anjolina (1994) and Ruy & Anjolina (2011): Term rewriting system –LNDEQ-TRS

• Total of 39 reduction rules – 7 essential to the current work

Reductions Involving Symmetry andReflexivity

• Obtained rules:

Reductions Involving Transitivity

• Obtained rules:

Reductions Involving Transitivity

• Obtained rule:

rw-equality

• Each LNDEQ-TRS is known as rw – rule

• From s to t in 1 rule:

• From s to t in multiple rules:

• Rw-equality s =rw t: sequence R0, ....., Rn , with such that:

• Rw-equality is an equivalence class (since it has been defined as a transitive, symmetric and reflexive closure)

LNDRW-TRS2 – Redundancies between Paths ofPaths• Redundancies caused by rw-equality

• There is a version for each redunction previously showed

• Exemple:

rw2-equality

• Rw2-equality: similar to rw-equality

• Rw2 is an equivalence class (analogous to rw)

• Special rule cd2:

Category Arw Induced by Computational Paths

• Objects: terms a: A

• Morphisms: Paths s between objects a,b: A. iff a =s b

• Composition:

• Identity:

• Weak category: Equality holds only up to rw-equality

• Associativity:

• Identity Laws:

ARW is a Weak Groupoid

• Every arrow is an isomorphism

• We need to show that every morphism s has an inverse morphism t

• Set :

Higher Strucure: 2 - Arw

• Category A2rw(a,b) for each pair of objects Arw

• Objects of A2rw(a,b) are paths s: a =s b and morphisms between paths s,r are sets of rw-equalities s =rw r

• Associativity and transitivity hold weakly up to rw2-equality(analogous to Arw)

• Considering equivalence classes of rw2, equalities hold “on the nose” on the second level. Structure [2 – Arw]

• Is [2 – Arw] a bicategory? Is it a weak 2-grupoid?

Bicategory

• Horizontal Composition

• Associativity and identity of the horizontal composition

• Interchange law

• Coherence law: Mac Lane’s pentagon and triangle

[2 – Arw] is a Bicategory

• Horizontal composition :

Given:

We define as:

[2 – Arw] is a Bicategory

• Associativity assoc of : Natural isomorphism between

Given by the isomorphism of each component:

• Identity :

We only need to check each component:

Analogous to : Use trr

[2 – Arw] is a Bicategory

• Interchange law:

[2 – Arw] is a Bicategory

• Coherence laws:

[2 – Arw]: Results

• We have showed that [2 – Arw] is a bicategory

• From the fact that [2 – Arw] is a bicategory, that A2rw is a weakgroupoid and Arw a groupoid, we conclude that [2 – Arw] is a weak 2-groupoid

• It is possible to think of weak groupoids with higher number of levels. Eventually, we can think of a weak groupoid with an infinite numberof levels, the weak

Conclusion

• We have proposed a new approach based on computer

• Computer paths are present in the syntax of type theory, opposite tobeing only a semantical interpretation

• Using computational paths, it is possible to induce higher groupoids.

• We have obtained results compatible with the ones obtained byHofmann-Streicher for the traditional identity type

Future Work

• Mapping of all possible rw2-rules

• The study of induced weak groupoids with order higher than 2

• Possibility of obtaining, to our approach based on computationalpaths, results similar to the ones obtained by Lumsdaine. In otherwords, to prove that it is possible to induce a