Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on commutative subrings of $R$.
This definition is motivated by quantum mechanics; roughly speaking $\phi$ preserves what a classical observer can observe about the noncommutative spaces $\text{Spec } R$ and $\text{Spec } S$. See the discussion at the nLab page on the Bohr topos. Actually it should be more like this:
Definition: Let $R, S$ be two $^\ast$-rings. A classical morphism $\phi : R \to S$ is a function from normal elements of $R$ (elements such that $r^{\ast} r = r r^{\ast}$) to normal elements of $S$ which restricts to a $^{\ast}$-homomorphism on commutative $^{\ast}$-subrings of $R$.
This definition allows, among other things, an elegant statement of the Kochen-Specker theorem, which can be restated as the claim that if $H$ is a Hilbert space of dimension at least $3$, then the algebra $B(H)$ of bounded linear operators $H \to H$ does not admit a classical morphism to $\mathbb{C}$.
Has this definition been studied from a purely ring theory or noncommutative geometry point of view? Have basic properties of the corresponding category been worked out somewhere?