In his 1973 topos seminar in Buffalo (the tapes are now freely available online!), Grothendieck said:
The cohomology of a topos associated to an algebraic structure should be called the "classifying cohomology" of this algebraic structure, which I think has significance.
In modern terminology: each geometric theory $\mathbb T$ induces a classifying topos $\mathcal E_\mathbb T$, whose cohomology groups $$H^n(\mathcal E_\mathbb T, A)$$ for abelian groups $A$ in $\mathcal E_\mathbb T$ might be interesting and tell us something about $\mathbb T$. Similarly for first-order theories.
In fact, in Pursuing stacks Grothendieck says the same (see page 7):
There is a general theorem for the existence of universal structures, covering all these cases - for instance there is a "classifying topos" for most algebro-geometric structures, whose cohomology say should be viewed as the "classifying cohomology" of the structure species considered.
Question: Has someone thought or written about this? Could that have applications in model theory?
