The real spectrum functor is an analog of Spec for partially ordered commutative rings and real closed fields in place of commutative reals and algebraically closed fields. I was hoping that there would also be a nice Spec-like functor for not-necessarily-commutative rings, but this paper seems fairly discouraging about that. Are there other functors that people have studied which are like Spec but from other categories (particularly varieties of algebras, or the category of models of a first-order theory)? More generally, what properties should a functor satisfy for us to consider it Spec-like, and what can then be said about Spec-like functors in general?

Stone Spaces. $\endgroup$somecategory of "spaces", one answer is that the dual of a category of ring-like objects can often be regarded as a category of "spaces" directly (e.g. affine schemes)-- so that the identity functor is "Spec-like". For example, if $\mathcal{V}$ is a Cauchy complete symmetric monoidal semiadditive category, then the opposite of the category of monoids in $\mathcal{V}$ isextensive, so "space-like" in at least a weak sense. $\endgroup$2more comments