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.

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