  We study the representation theory of the Lie superalgebra
$\mathfrak{gl}(1|1)$, constructing two spectral sequences which eventually
annihilate precisely the superdimension zero indecomposable modules in the
finite-dimensional category. The pages of these spectral sequences, along with
their limits, define symmetric monoidal functors on $\mathrm{Rep}
(\mathfrak{gl}(1|1))$. These two spectral sequences are related by
contragredient duality, and from their limits we construct explicit
semisimplification functors, which we explicitly prove are isomorphic up to a
twist. We use these tools to prove branching results for the restriction of
simple modules over Kac-Moody and queer Lie superalgebras to
$\mathfrak{gl}(1|1)$-subalgebras.
