  We prove a general result that relates certain pushouts of $E_k$-algebras to
relative tensors over $E_{k+1}$-algebras. Specializations include a number of
established results on classifying spaces, resolutions of modules, and
(co)homology theories for ring spectra. The main results apply when the
category in question has centralizers.
  Among our applications, we show that certain quotients of the dual Steenrod
algebra are realized as associative algebras over $HF_p \wedge HF_p$ by
attaching single $E_1$-algebra relation, generalizing previous work at the
prime $2$. We also construct a filtered $E_2$-algebra structure on the sphere
spectrum, and the resulting spectral sequence for the stable homotopy groups of
spheres has $E_1$-term isomorphic to a regrading of the $E_1$-term of the May
spectral sequence.
