[too speculative for MO] is there any derived lift of the theory of unstable modules for the Steenrod algebra? is there any known analogue for extraordinary cohomology theories? would love to be able to say that the unstable algebras live in something like the heart of a t-structure, though I doubt that is literally true

maybe better to ask the more basic question: how to think about instability of modules other than as a condition which happens to hold in the cohomology of a space?