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From: Stefan Forcey <sforcey@math.vt.edu>
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Subject: many object version of promonoidal category?
Date: Mon, 17 Nov 2003 14:22:32 -0500 (EST)
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Hello,
 In the following reference

[1]  B.J. Day, On closed categories of functors, Lecture Notes in
Math 137 (Springer, 1970) 1-38

are defined promonoidal, or monoidal enriched categories. It seems that
there should be some well known many object version of this, in the sense
that a bicategory is the many object version of a monoidal category. Does
anyone know a definition or, even better, a reference?
 A much later related definition is in the appendix of

[2] V. Lyubashenko, Category of $A_{\infty}$--categories,
    Homology, Homotopy and Applications 5(1) (2003), 1-48.

Here are defined enriched 2-categories. This seems to be the strict case
of what I'm looking for, since a promonoidal category is a monoid in the
category of enriched categories, or a one-object category enriched over
V-Cat. In [2] enriched 2-categories are defined as enriched over V-Cat.

Thanks,
 Stefan Forcey