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