Homotopical algebraic context are models that allows Toën and Vezzosi to do derived geometry. It have been defined in their seminal paper Homotopical Algebraic Geometry II.
These are general abstract contexts where you can see functions on affines to be modeled on object on some abstract symmetric monoidal category satisfying some assumptions (base change especially).
I don't recall Lurie investigating it but I believe that spectral algebraic geometry (connective or non-connective) can be defined in this setting. In HAG II, Toën and Vezzosi define such contexts and define geometry on it but they don't really set up "morphisms of contexts" in order to compare different geometries coming from different contexts (an example of such comparison can be found in a classical setting in Au-dessous de Spec(Z) by Toën and Vaquié).
Has a theory of contexts has been investigated so far ? I am especially interested in comparsion between the theories of quasi-coherent sheaves and comparision between geometricity in two different contexts as it seems an interesting problem.
