Operads, configuration spaces and quantization

We review several well-known operads of compactified configuration spaces and construct several new such operads, $\overline{C}$,
in the category of smooth manifolds with corners
whose complexes of fundamental chains give us
(i) the 2-coloured operad of $A_\infty$-algebras and their homotopy morphisms, (ii)
the 2-coloured operad of $L_\infty$-algebras and their homotopy morphisms, and (iii)
the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms.

Two gadgets --- a (coloured) operad of Feynman graphs and a de Rham field theory
on $\overline{C}$ --- are introduced and used to construct quantized representations
of the (fundamental)
chain
operad of
$\overline{C}$ which are given by Feynman type sums over graphs and depend on choices of
propagators.