Coherence via Wellfoundedness
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Homotopy type theory allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory. The notion of coherence is central: it is usually not enough to know that objects are equal, but one needs to know how they are equal, and collections of equality proofs need to fit together. 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). This makes it hard to characterise the type of maps from a quotient into a higher type. Open questions about free higher groups, pushouts of sets, and a well-behaved type-theoretic representation of type theory itself emerge from this difficulty. In order to approach these problems, we work with cycles (closed zig-zags) given by the transitive closure of the relation. Reasoning about these is hard since the obvious inductive strategy is bound to fail: if we remove a segment from a closed zig-zag, it is not closed anymore. We make use of the observation that the relations in the mentioned problems are all confluent and wellfounded. Starting from such a relation, we construct a new relation on closed zig-zags with the same properties, making wellfounded induction possible. This allows us to approach approximations (1-truncated cases) of the mentioned open questions. We have formalised our theory of cycles over confluent and well-founded relations, and the derivation of our main theorem, in the proof assistant Lean.
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