  In this survey article we discuss certain homotopy coherent enhancements of
the coalgebra structure on cellular chains defined by an approximation to the
diagonal. Over the rational numbers, $C_\infty$-coalgebra structures control
the $\mathbb Q$-complete homotopy theory of spaces, and over the integers,
$E_\infty$-coalgebras provide an appropriate setting to model the full homotopy
category. Effective constructions of these structures, the focus of this work,
carry geometric and combinatorial information which has found applications in
various fields including deformation theory, higher category theory, and
condensed matter physics.
