Abstract
We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.
Highlights
We investigate types of infinite data with binding
We show that the syntactic approach of completing and quotienting agrees with the semantic approach of final coalgebras in nominal sets
We describe the corecursion principle ensuing from final coalgebras in nominal sets and show that (1.3) is a coinductive definition of substitution for infinitary λ-terms
Summary
We investigate types of infinite data with binding. A leading example is the infinitary λcalculus. The vertical map in the middle column Λ∞ → (Λ/=α)∞ is not onto (Example 5.20), metric completion and quotienting by α-equivalence do not commute for terms with countably many free variables (as opposed to the case of uncountably many variables, see Theorem 5.19). The vertical map Λ∞ ffv → (Λ/=α)∞ fs in the right-hand column is onto [KPSdV12, Theorem 22], in other words, restricted to terms with finitely many free variables, the two operations of metric completion and quotienting by α-equivalence do commute.
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