Date: Wed, 24 Jul 1996 10:44:40 -0300 (ADT)
Subject: Re: lifting of sm(c)-structure

Date: Fri, 19 Jul 1996 10:22:31 +1000
From: Ross Street <street@mpce.mq.edu.au>

Answer to the following edited question:

>Date: Wed, 17 Jul 1996 18:18:22 MESZ
>From: Thomas Streicher <streicher@mathematik.th-darmstadt.de>
>To: categories@mta.ca
>Subject: lifting of sm(c)-structure

>could someone provide me with a reference of a paper of B. Day where he
>described how to lift a symmetric monoidal (closed) structure on a small
>category C to the category of presheaves C^ = Set^C^op in a way that the
>Yoneda functor preserves the symmetric monoidal structure ??

>Thomas Streicher

This was done in Brian Day's thesis [D1]. It is accessible in [D2], [D3].
Starting with any monoidal category C with tensor product #, we obtain a
promonoidal category C with P(a,b,c) = C(a#b,c). Then the presheaf category
C^ becomes a (left and right) closed cocomplete monoidal category under the
convolution tensor product; the Yoneda embedding y : C --> C^ is strong
monoidal (preserves tensor). In fact, this is the universal monoidal
cocompletion (see [IK]). If C is symmetric, so is C^ and y preserves
the symmetry.

[D1]    B.J. Day, Construction of Biclosed Categories, PhD Thesis,
University of New South Wales, Australia (1970).

[D2]    B.J. Day, On closed categories of functors, Midwest Category
Seminar Reports IV, Lecture Notes in Math. 137 (Springer, 1970) 1-38.

[D3]    B.J. Day, An embedding theorem for closed categories, Category
Seminar Sydney 1972-73, Lecture Notes in Math. 420 (Springer, 1970) 55-64.

[D4]    B.J. Day, An embedding of bicategories, (Dept of Pure Math Report,
The University of Sydney, 1976)

[D5]    B.J. Day, Note on monoidal monads, Journal of the Australian Math
Society 23 (1977) 292-311.

[D6]    B.J. Day,  Promonoidal functor categories, Journal of the
Australian Math Society 23 (1977) 312-328.

[D7]    B.J. Day, Biclosed bicategories: localisation of convolution,
Macquarie Math Reports  #81-0030 (April 1981).

[IK]    G.B. Im and G.M. Kelly, A universal property of the convolution
monoidal
 structure,  Journal of Pure & Applied Algebra 43 (1986) 75-88.


Best regards,
Ross (Street)