I've just beginning to learn logic and proof theory - and the following rather vague and perhaps ill-formed question occurred to me.
Given an alphabet it's straightforward to construct the Language, that is the monoid of all sentences (by concatanation as the operation and using the empty sentence as the unit).
A particular grammar for that alphabet will select a sublanguage.
Is it possible to turn into a Galois connection? So a morphism between Grammars would give one between Languages etc. Of course this would mean we must a robust description as to what is meant by a Grammar.
Has anyone done work in this direction, or is this angle simply not formalisble in any sensible or useful way or is it actually a re-capitulation of Lawveres Galois connection in Model theory between Theories and Models?