  We start by clarifying and extending the multibraces notation, which
economically describes substitutions of multilinear maps and tensor products of
vectors. We give definitions and examples of homotopy algebras, strongly
homotopy Gerstenhaber and Gerstenhaber bracket algebras, and strongly homotopy
Batalin-Vilkovisky algebras. We show that a homotopy algebra structure on a
vector space can be lifted to its Hochschild complex, and also suggest an
induction method to generate explicit strongly homotopy Gerstenhaber algebra
maps on a topological vertex operator algebra (TVOA), their existence having
been proven by Kimura, Voronov, and Zuckerman in 1996 (later amended by
Voronov). The contention that this is the fundamental structure on a TVOA is
substantiated by providing an annotated dictionary of strongly homotopy BV
algebra maps and identities found by Lian and Zuckerman in 1993.
