categories: Re: Operads

["Martin Markl, Mathematical Institute of the Academy" <markl@matsrv.math.cas.cz>]

   Peter May asked me to send in a correction of my message enclosed
below;  he says that PROPs were invented in unpublished joint work of
Adams and Mac Lane, not Mac Lane alone.


---------- Forwarded message ----------
Date: Mon, 28 Jan 2002 08:46:29 +0100 (CET)
From: "Martin Markl, Mathematical Institute of the Academy"
    <markl@matsrv.math.cas.cz>
To: categories@mta.ca, S.J.Vickers@open.ac.uk
Cc: Steve Shnider <shnider@macs.biu.ac.il>,
     JAMES STASHEFF <stasheff@email.unc.edu>
Subject: Re: categories: Operads (fwd)

   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
> 
> 
> 
> 
>