2 research outputs found
W-types in setoids
W-types and their categorical analogue, initial algebras for polynomial
endofunctors, are an important tool in predicative systems to replace
transfinite recursion on well-orderings. Current arguments to obtain W-types in
quotient completions rely on assumptions, like Uniqueness of Identity Proofs,
or on constructions that involve recursion into a universe, that limit their
applicability to a specific setting. We present an argument, verified in Coq,
that instead uses dependent W-types in the underlying type theory to construct
W-types in the setoid model. The immediate advantage is to have a proof more
type-theoretic in flavour, which directly uses recursion on the underlying
W-type to prove initiality. Furthermore, taking place in intensional type
theory and not requiring any recursion into a universe, it may be generalised
to various categorical quotient completions, with the aim of finding a uniform
construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio
W-types in setoids
We present a construction of W-types in the setoid model of extensional
Martin-L\"of type theory using dependent W-types in the underlying intensional
theory. More precisely, we prove that the internal category of setoids has
initial algebras for polynomial endofunctors. In particular, we characterise
the setoid of algebra morphisms from the initial algebra to a given algebra as
a setoid on a dependent W-type. We conclude by discussing the case of free
setoids. We work in a fully intensional theory and, in fact, we assume identity
types only when discussing free setoids. By using dependent W-types we can also
avoid elimination into a type universe. The results have been verified in Coq
and a formalisation is available on the author's GitHub page