Internal language [mde-0027]

For some syntax, given for instance by a type theory, string diagrams, etc. we say it forms the internal language of some class of structured algebraic gadgets (such as particular doctrine of categories) if the syntactic model (over some signature) forms the free such gadget.

An internal language is necessary sound and complete as an equational theory for the class of algebraic gadgets.

Completeness is the statement that if an equation holds in all models, it holds in the syntax. Since our syntax here is just the free model, completeness holds trivially: a formula holding in all models holds in the free model in particular.

Soundness is the statement that any formula holding in the syntactic model (over a signature) holds in every model. Let some interpretation of a signature \(S\) be given in a model \(M\), then from freeness we have a unique morphism from the syntax freely generated by \(S\), into \(M\). In particular, equal terms in the syntactic theory map to equal terms in the model.