Dear Steve Vickers PROPs invented by Mac Lane in the 1960 do the job; operad is then a special case of a PROP. For example, Hopf algebras and various other bialgebras are described by PROPs. I believe you can find more details in @article{maclane:BAMS65, author={S. {Mac~Lane}}, title={Categorical Algebra}, journal={Bull. Amer. Math. Soc.}, year=1965, volume=71, pages={40--106}} There is also a more special notion called bioperad introduced recently by Wee Liang Gan - see his paper math.QA/0201074 posted on xxx.lanl.gov. Sincerely, Martin
Date: Mon, 19 Nov 2001 09:56:13 -0000 From: S.J.Vickers@open.ac.uk To: categories@mta.ca Cc: univalg@yahoogroups.com Subject: categories: Operads
There's some discussion on the Universal Algebra list at present on operads.
I'm not very familiar with them. What I understand from the discussion is they capture single sorted algebraic theories with respect to a symmetric monoidal product ox. For each natural number n an object of n-ary operators O_n is given, and an algebra A has operations O_n ox A^(n) -> A where A^(n) is A ox ... ox A n times.
If you do this sort of thing with respect to categorical product, then it already contains the information of the Lawvere theory category (for single-sorted finitary algebraic theories), since hom(m,n) is hom(m,1)^n and you take hom(m,1) to be O_m. But with a monoidal product this doesn't work. It seemed to me that for proper generality the operad ought to have objects O_mn (m, n natural numbers) representing the object of operations from A^(m) to A^(n). Is there a name for that?
Steve Vickers Department of Pure Maths Faculty of Maths and Computing The Open University ----------- Tel: 01908-653144 Fax: 01908-652140 Web: http://mcs.open.ac.uk/sjv22
29-Jan-2002 20:56:48 -0400,2378;000000000000-00000000