tensorial strength

In reply to Jeff Egger: Actions by monoidal categories were considered by Benabou in 1967 ("Intoduction to bicategories", Midwest Category Seminar I); strength of functors between categories on which a monoidal category acts were considered by H. Lindner, in some papers/preprints in the late 70s, and a summary is given in his "Enriched categories and enriched modules", Cahiers Vol 22 (1981), 161-174. This paper also contains several references. -Aspects of tensorial strength has been developed by several computer-science mathematicians later (starting with E. Moggi, I believe); I do not know much about their work, so my answer to Jeff's question may not be up to date. Anders Kock