Coherence via Well-Foundedness

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 wellbehaved, 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 1types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean.

On homotopy of walks and spherical maps in homotopy type theory

“…On the reduction relation on walks and spherical maps, this work is related to polygraphs used in the context of higher-dimensional rewriting systems. Recent works by Kraus and von Raumer [13,14] use ideas in graph theory, higher categories, and abstract rewriting systems to approximate a series of open problems in HoTT. In the same vein, the internalisation of rewriting systems and the implementation of polygraphs in Coq by Lucas [15,16] was found to be related to Kraus and von Raumer's approach.…”

“…However, to the best of our knowledge, few efforts use a proof-relevant dependent type theory like HoTT and a proof assistant like Agda. We find only the work mentioned earlier by Kraus and von Raumer [13,14] to be related to our Agda development; their work contains a formalisation of their results in a version of the proof assistant Lean compatible with HoTT.…”

Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

“…Of course, the intermediate goal that this paper is working towards is the statement that 2LTT can eat HoTT. Even a much weaker question, namely whether the initial ∞-CwF has trivial fundamental group, would be very interesting; but already this seemingly much simpler problem appears to be highly non-trivial, and similar results for seemingly simpler situations [65]- [67] suggest that a solution will require new techniques.…”