What is the Spectral Category?

For a category with finite limits and a class of monomorphisms in that is pullback stable, contains all isomorphisms, is closed under composition, and has the strong left cancellation property, we use pullback stable -essential monomorphisms in to construct a spectral category. We show that it has finite limits and that the canonical functor (formula presented) preserves finite limits. When (formula presented) is a normal category, assuming for simplicity that (formula presented) is the class of all monomorphisms in (formula presented), we show that pullback stable (formula presented)-essential monomorphisms are the same as what we call subobject-essential monomorphisms.