Who first defined categories enriched over a _promonoidal_ base?

Dear Robin, I don't think I can directly answer your question; if it has a direct answer I would very much like to know it as well. However, I have spent some time thinking about this and related questions and know of some resources that might help. Promonoidal categories over a closed symmetric monoidal V are monoids in V-Mod, and thus a promonoidal structure on a V-category A is a V-functor P:A^op \otimes A^op \otimes A -> V. ( My favourite source is section 7 of the paper by Brian Day and Ross Street. Monoidal Bicategories and Hopf Algebroids; advances in mathematics 129, 99-157 (1997)) Unlike for a monoidal V-category (monoid in V-Cat) this does not as far as I can see immediately provide a monoidal structure on the underlying category of A. But there are two indirect possibilities for the meaning of enrichment over a promonoidal base. One is to enrich over the functor category [A,V] , which is automatically monoidal. See the original paper by Brian Day. { B.J. Day, On closed categories of functors, Lecture Notes in Math 137 (Springer, 1970) 1-38.} The other is in the special case of a closed promonoidal category in which we have an additional functor *:A \otimes A^op -> A for which P(a,b,c) = A(b, a*c) the latter being the appropriate hom-object of A in V. Thus we could conceivably define a category B enriched over A as having hom objects B(x,y) in A and composition morphisms in V given by M:I-> P(B(y,z), B(x,y), B(x,z)). This should reduce to being the same as enriching over the underlying category of A, but I have yet to straighten out the details, especially the correct locations of ^op! Again, if someone else has done so I would like to see the source as well. (The idea of (bi)closed promonoidal is introduced by Day in the context of probicategories in a preprint: "Biclosed bicategories: localization of convolution" in the Maquarie Mathematics Reports, which really ought to be published at some point.) Of further interest is the generalization of promonoidal categories in the form of substitudes, which are also a little more general than multicategories. See the preprint of Street and Day: (available from Ross's website) Abstract substitution in enriched categories; Brian Day and Ross Street. Whether there is a theory of enrichment over these very appealing objects is a question for the authors. Perhaps there is an additional enrichment facilitating requirement on substitudes which generalizes the closed promonoidal concept and/or is along the lines of Tom Leinster's generalized enrichment over T-multicategories (see his papers on the arxiv.) Hope some of this helps, Stefan Forcey Robin Houston writes:

Dear Categorists,

Who first defined categories enriched over a _promonoidal_ base?

I'd particularly like references, but any information would be welcome.

Robin

29-Mar-2005 12:56:27 -0400,2742;000000000001-00000000