monoidal terminology

monoidal terminology

[jean benabou]

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

Many thanks, Jean Benabou.

Re: monoidal terminology

[Ross Street]

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