[Paper Review] Coherence via Well-Foundedness: Taming Set-Quotients in Homotopy Type Theory

Nicolai Kraus, Jakob von Raumer|arXiv (Cornell University)|Jan 21, 2020

Homotopy and Cohomology in Algebraic Topology15 references5 citations

TL;DR

This paper introduces a novel induction principle for cycles in graphs defined by a locally confluent and well-founded relation, enabling coherence proofs for set-quotients in homotopy type theory. It establishes that coherence for cycles reduces to verifying base cases and local confluence, enabling the construction of functions from set-quotients into 1-types and proving that the second homotopy group of free higher groups is trivial.

ABSTRACT

Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence. Our motivation and application is in the field of homotopy type theory, which allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory, making coherence a central issue. This is in particular true for quotienting - a natural operation which gives a new type for any binary relation on a type and, in order to be well-behaved, cuts off higher structure (set-truncates). The latter makes it hard to characterise the type of maps from a quotient into a higher type, and several open problems stem from this difficulty. We prove our theorem on cycles in a type-theoretic setting and use it to show coherence conditions necessary to eliminate from set-quotients into 1-types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean.

Motivation & Objective

• To address the coherence problem in homotopy type theory when mapping set-quotients into higher types.
• To overcome the challenge that cycles (closed zig-zags) are not inductively generable from non-closed zig-zags.
• To develop a usable induction principle for cycles under well-founded and locally confluent relations.
• To apply this principle to prove that the second homotopy group of the free higher group over a set is trivial.
• To formalize the main results in the Lean proof assistant for verification.

Proposed method

• Introduces a Noetherian cycle induction principle based on well-foundedness and local confluence.
• Defines a canonical map from the untruncated quotient (coequaliser) to the free higher group.
• Uses the coherence condition from Theorem 13: a cycle maps to a trivial loop in the target 1-type.
• Applies the induction principle to prove that all cycles in the pushout of a span of 1-types become trivial.
• Leverages the fact that lists over sets are themselves sets, enabling use of simpler truncation theorems.
• Formalizes the core result in Lean, ensuring correctness and usability.

Experimental results

Research questions

• RQ1 Can we construct a usable induction principle for cycles in graphs defined by a well-founded, locally confluent relation?
• RQ2 Is it possible to prove that all cycles in the free higher group over a set become trivial, without assuming decidable equality?
• RQ3 Can we derive a general coherence condition for functions from set-quotients into 1-types?
• RQ4 Does the Noetherian cycle induction principle allow us to bypass the need for explicit inverse constructions in homotopy type theory?
• RQ5 Can this method be extended to prove triviality of higher homotopy groups beyond π₂?

Key findings

• The paper proves that ∥Ω²(B +ₐ C, i(x))∥₁ is a set, implying π₂(B +ₐ C) = 0 for 1-types B, C and a set A.
• The second homotopy group of the free higher group over a set is trivial, confirming a long-standing open problem in HoTT.
• The coherence condition in Theorem 13 is sufficient to construct functions from set-quotients into 1-types when cycles are trivialized.
• The main result is formalized in Lean, ensuring its correctness and applicability in proof assistants.
• The method applies to pushouts and free groups, showing that coherence reduces to checking local confluence and base cases.
• The conditions in Theorem 50 are shown to be maximally general, as relaxing them leads to counterexamples.