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 Answer to the following edited question: >Date: Wed, 17 Jul 1996 18:18:22 MESZ >From: Thomas Streicher >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)