Initial Semantics for Strengthened Signatures

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].