Dear Jean
Suppose V is a (fixed) monoidal symmetric category and C is a category enriched over V .
The following notions should be "obviously well-known" , but I cannot find any reference for them in the "standard literature" , do such references exist ?
1- A monoidal structure on C (of course, as an enriched category) 2- A symmetric monoidal structure on C 3- A closed monoidal structure on C
As you expected these concepts are truly well known. I can give two references that I would consider part of the "standard literature", at the two ends of a chronological spectrum: [1] B.J. Day, On closed categories of functors, Lecture Notes in Math 137 (Springer, 1970) 1-38 [2] B.J. Day, P. McCrudden and R. Street, Dualizations and antipodes, Applied Categorical Structures 11 (2003) 229-260 In [1] you will find the notion of a "premonoidal V-category" A which, because of your term "profunctor", was renamed "promonoidal V-category". This paper contains part of Brian Day's PhD thesis. It carefully explains explicitly how a monoidal V-category is promonoidal, and what it means for it to be closed. It also carefully defines symmetry for promonoidal V-categories (and shows how it amounts to a symmetry for the convolution monoidal structure on the V-category [A,V] of V-functors A --> V). In [2] you will find monoidal (or pseudomonoid) structures, together with closed, symmetric and braided ones, on objects in any autonomous monoidal bicategory (such as V-Mod, V-Prof or V-Dist, whichever name you prefer). For the three matters in question, this is perhaps an improvement on B.J. Day and R. Street, Monoidal bicategories and Hopf algebroids, Advances in Math. 129 (1997) 99-15. Best regards, Ross