The classical associahedra are cell complexes, in fact polytopes,
introduced by Stasheff to parametrize the multivariate operations
naturally occurring on loop spaces of connected spaces.
They form a topological operad $ Ass_\infty $ (which provides a resolution
of the operad $ Ass $ governing spaces-with-associative-multiplication)
and the complexes of cellular chains on the associahedra form a dg
operad governing $A_\infty$-algebras (that is, a resolution of the
operad governing associative algebras).
In classical applications it was not necessary to consider units for
multiplication, or it was assumed units were strict. The introduction
of non-strict units into the picture was considerably harder:
Fukaya-Ono-Oh-Ohta introduced homotopy units for $A_\infty$-algebras in
their work on Lagrangian intersection Floer theory, and equivalent
descriptions of the dg operad for homotopy unital $A_\infty$-algebras
have now been given, for example, by Lyubashenko and by Milles-Hirsch.
In this talk we present the "missing link": a cellular topological
operad $uAss_\infty$ of "unital associahedra", providing a resolution
for the operad governing topological monoids, such that the cellular
chains on $uAss_\infty$ is precisely the dg operad of
Fukaya-Ono-Oh-Ohta.
(joint work with Fernando Muro, arxiv:1110.1959, to appear Forum Math)