André Joyal wrote:
I wonder who first introduced the notion of bilax monoidal functor and when?
I believe that Aguiar and Mahajan were the first to formally introduce this concept, though the Alexander-Whitney-Eilenberg-MacLane example has been around for a long time. On the n-Category Cafe, Kathryn Hess recently wrote:
The A-W/E-Z equivalences for the normalized chains functor are a special case of the strong deformation retract of chain complexes that was constructed by Eilenberg and MacLane in their 1954 Annals paper "On the groups H(π,n). II". For any commutative ring R, they defined chain equivalences between the tensor product of the normalized chains on two simplicial R-modules and the normalized chains on their levelwise tensor product.
Steve Lack and I observed recently that the normalized chains functor is actually even Frobenius monoidal. We then discovered that Aguiar and Mahajan already had a proof of this fact in their recent monograph. :-)
I forget if "Frobenius monoidal" is a precise synonym of "bilax monoidal". Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]