many object version of promonoidal category?

many object version of promonoidal category?

[Stefan Forcey]

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

Re: many object version of promonoidal category?

[Ross Street]

Dear Stefan

>[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?

Brian Day put out a short preprint:

	Brian J. Day, Biclosed bicategories: localisation of convolution,
	Macquarie Mathematics Reports #81-0030 (April 1981)

but it was (allegedly) too far ahead of its time to be published.
Here "biclosed" means that all right extensions and right liftings
exist. So a one-object "biclosed bicategory" is a monoidal category
with both left and right internal homs.

In the short paper he defines what I think is exactly what you want
and calls them "probicategories".  By performing convolution on the
homs one obtains biclosed bicategory which is locally cocomplete.

In more recent work, Brian and I have found something more general
than probicategories to be useful. Again, afraid of going too
general, we have concentrated on the one object case; thus we have
things called "substitudes" which are lax versions of promonoidal
V-categories. They also generalise Lambek's multicategories. For the
promonoidal case of a substitude, the multihoms are all determined up
to canonical isomorphism by the nullary, unary and binary homs. See
for example:

72. (with B.J. Day) Lax monoids, pseudo-operads, and convolution, in:
"Diagrammatic Morphisms and Applications", Contemporary Mathematics
318 (AMS; ISBN 0-8218-2794-4; April 2003) 75-96.

77. (with B.J. Day) Abstract substitution in enriched categories, J.
Pure Appl. Algebra 179 (2003) 49-63.

The natural level of generality for the subject of

70. (with G.M. Kelly, A. Labella and V. Schmitt) Categories enriched
on two sides, J. Pure Appl. Algebra 168 (1) (8 March 2002) 53-98

seems to be substitudes-with-several-objects rather than
bicategories. I actually wrote some draft sections on that during the
writing of [70] but we chickened out.

Best wishes,
Ross