Initial Semantics for Strengthened Signatures

Abstract

We give a new general definition of arity, yielding the companion notions of signature and associated syntax. This setting is modular in the sense requested by [12]: merging two extensions of syntax corresponds to building an amalgamated sum. These signatures are too general in the sense that we are not able to prove the existence of an associated syntax in this general context. So we have to select arities and signatures for which there exists the desired initial monad. For this, we follow a track opened by Matthes and Uustalu [16]: we introduce a notion of strengthened arity and prove that the corresponding signatures have initial semantics (i.e. associated syntax). Our strengthened arities admit colimits, which allows the treatment of the λ-calculus with explicit substitution in the spirit of [12].

Heterogeneous substitution systems revisited

“…The notion of signature introduced in [23] and formalized in the present work is similar to that employed in Hirschowitz and Maggesi's most recent work [18]. One difference is that we do not, in the present work, insist on our signature functor to be ω-cocontinuous, since we do not worry about the existence of initial algebras, but assume them to exist.…”

Bindings as bounded natural functors

“…Recent work [Ahrens et al 2018] on the univalent foundation of mathematics (having roots in earlier category-theoretic advances [Ghani et al 2006;Hirschowitz and Maggesi 2012]) defines a general notion of signature for syntax with bindings and identifies criteria for the existence of initial models, again following the nameless paradigm. Like our work, it emphasizes modularity and studies the existence of substitution operators.…”