Date: Wed, 26 Nov 1997 13:44:04 -0400 (AST) Subject: Algebraic Theories/Operads Date: Wed, 26 Nov 1997 08:57:36 -0600 (CST) From: David Metzler I'm looking for a good reference on the relation between algebraic theories (a la Lawvere) and operads. David Metzler Date: Thu, 27 Nov 1997 16:04:58 -0400 (AST) Subject: RE: Algebraic Theories/Operads Date: Thu, 27 Nov 1997 16:10:31 MET From: Heinrich.Kleisli@unifr.ch May, J. P.: Operads, algebras and modules. Contemp. Math. 202, 15-31 (1997). Date: Thu, 27 Nov 1997 16:03:21 -0400 (AST) Subject: Re: Algebraic Theories/Operads Date: Wed, 26 Nov 1997 16:22:14 -0800 (PST) From: john baez David Metzler writes: > I'm looking for a good reference on the relation between > algebraic theories (a la Lawvere) and operads. Me too! The closest thing I know is Boardman and Vogt's book on homotopy invariant algebraic structures --- which uses the framework of PROPs rather than operads. Date: Thu, 27 Nov 1997 16:04:13 -0400 (AST) Subject: Re: Algebraic Theories/Operads Date: Thu, 27 Nov 1997 10:40 +0530 From: CAYLEY@tifrvax.tifr.res.in The volume 202 of Contemprory mathematics series of the AMS titled Operads: Proc. of Renaissance conferences ed by J.Loday et al is a possible source. P.S.Subramanian, Tata Institute. Date: Sat, 6 Dec 1997 16:12:19 -0400 (AST) Subject: Re: Algebraic Theories/Operads Date: Fri, 5 Dec 1997 18:54:36 +0000 (GMT) From: Tom Leinster > > I'm looking for a good reference on the relation between > algebraic theories (a la Lawvere) and operads. > > David Metzler I think the relationship between algebraic theories and (non-symmetric, non-topological) operads is quite simply described. Loosely, algebras for operads are just the same as algebras for strongly regular theories. To be more precise: any operad gives rise to a monad on Set, the algebras for which are the algebras for the operad; a monad on Set arises from an operad iff it arises from a strongly regular theory. So - operadic monads are strongly regular theories. Carboni & Johnstone (1995) call an equation _strongly regular_ if on each side the same variables appear in the same order, without repetition (e.g. (x.y).z=x.(y.z) and x-(y-z)=(x-y)+z, but not x.y=x, x.y=y.x or (x.x).y=x.(x.y)). A theory is called strongly regular if it can be presented by operators and strongly regular equations - for instance, the theory of monoids. They show that a monad (T, eta, mu) on Set is from a s.r. theory ("is s.r.") iff (i) T is finitary (ii) T preserves wide pullbacks & (iii) eta and mu are cartesian. (A wide pullback is a limit over a diagram (X_i --> X) where i ranges over some set I, e.g. if I=2 it's an ordinary pullback.) 1. Operadic => strongly regular If A is an operad, with the function A --> N={natural numbers}, then the functor part of the induced monad T on Set is defined by the pullback square T(X) ----> W(X) | | | | W(!) | | V V A ------> W(1)=N where X is a set and W (for Words) is the free-monoid monad. One can show (e.g. my (1997, sec 4.6)) that the unit and multiplication of T are cartesian. Moreover, one can also show that (a) if W preserves colimits of a given shape then so does T & (b) if W preserves I-ary pullbacks then so does T. Since the theory of monoids is s.r., W preserves all filtered colimits (i.e. is finitary) and all wide pullbacks. So T satisfies (i)-(iii) and is therefore s.r.. 2. Strongly regular => operadic Conversely, take a s.r. theory T. Any s.r. presentation of T gives rise to a natural transformation T --> W which is cartesian and preserves the monad structure. It follows by my (1997, sec 4.6) that the monad T comes from some operad A. References: A Carboni, P T Johnstone (1995), Connected limits, familial representability and Artin glueing. Math Struct in Comp Science, vol 5, pp 441-459. T Leinster (1997, updated 3 Dec), General operads and multicategories. http://www.dpmms.cam.ac.uk/~leinster. I've heard tell that these ideas were explored by Kelly in his work on clubs - can anyone enlighten me? Tom Leinster Date: Mon, 8 Dec 1997 13:42:54 -0400 (AST) Subject: Re: Algebraic Theories/Operads Date: Mon, 8 Dec 1997 20:15:23 +1100 (EST) From: Barry Jay Tom, you may care to look at my paper "Languages for monoidal categories" @Article(Jay89b, Author={Jay, C.B.}, Title={Languages for monoidal categories}, Journal=jpaa, Volume=59, Year=1989, Pages={61--85}) and its successors (see my ftp site below). Barry Jay ************************************************************************* | Associate Professor C.Barry Jay, | | Reader in Computing Sciences Phone: (61 2) 9514 1814 | | Head, Algorithms and Languages Group, Fax: (61 2) 9514 1807 | | University of Technology, Sydney, e-mail: cbj@socs.uts.edu.au | | P.O. Box 123 Broadway, 2007, www: linus.socs.uts.edu.au/~cbj | | Australia. ftp: ftp.socs.uts.edu.au/Users/cbj | *************************************************************************