Abstract. We present a simple extension of the classical Hilton-Eckmann argument which proves that the endomorphism monoid of the unit object in a monoidal category is commutative. It allows us to recover in a uniform way well-known results on the graded-commutativity of cup products defined on the cohomology theories attached to various algebraic structures, as well as some more recent results.

We show that the action of the Lie algebra HH 1 (A) of outer derivations of an associative algebra A on the Hochschild cohomology HH• (A) of A given by the Gerstenhaber bracket can be computed in terms of an arbitrary projective resolution of A as an A-bimodule, without having recourse to comparison maps between the resolution and the bar resolution.In his classic paper On the cohomology structure of an associative ring [5], Murray Gerstenhaber introduced a Lie algebra structure on the Hochschild cohomology HH • (A) of an associative algebra A. This structure played a role in the proof contained in that paper of the commutativity of the cup product of HH • (A), he himself showed later in [6] that it is related to the deformation theory of A, and it has ever since been regarded as an important piece of the cohomological structure of the algebra. There has been a significant amount of effort expended by many authors in order to study this structure, specially in recent times.This Lie algebra structure on HH • (A) is defined in terms of a particular realization of Hochschild cohomology: the algebra A has a canonical bimodule bar resolution B(A) • , the Hochschild cohomology HH • (A) is canonically isomorphic to the cohomology of the complex hom A e (B (A) • , A), and the Lie bracket of HH • (A) is constructed using certain explicit formulas in terms of cochains in this complex. While this is convenient for many purposes, it is quite inconvenient in one important respect: we never compute Hochschild cohomology using the bar resolution. In practice, we pick a projective resolution P • of A which is better adapted to the task and compute instead the cohomology of the complex hom A e (P • , A), which is -thanks to the yoga of homological algebra-canonically *

Research partially supported by UBACYT TW69 and CONICET. 2 Research member of CONICET (Argentina). * (gr(B), gr(B)). The only thing to be checked now is that the differential on the E 1 term can be identified to .df , and this is easily done.