  We introduce an analogue of the famous Schur multiplier in the context of
associative trialgebras, or triassociative algebras. The latter were first
studied by Loday and Ronco in 2001, and are characterized by three operations
and eleven relations. The paper highlights an extension-theoretic crossroads of
multipliers, covers, and unicentral triassociative algebras. Using theory from
previous algebraic contexts as a guide, we develop criteria for when the center
of the cover maps onto the center of the algebra. Along the way, we obtain the
uniqueness of the cover, two different characterizations of the multiplier, and
several exact cohomological sequences.
