Let AA and BB be algebraic theories. The category [A,B][A,B] of (A,B)(A,B)-bimodels and their homomorphisms is the category of AA-models and homomorphisms in BMod opB Mod^{op}. An alternative description is that is a co-AA-model in BModB Mod. Each such bimodel MM determines and is determined by a pair of adjoint functors

Hom B(M,?):BModAModHom_B(M,?): B Mod \to A Mod
M A?:AModBModM\otimes_A ?: A Mod \to B Mod

Composition of such adjoint pairs yields a functor

B:[B,C]×[A,B][A,C]\otimes_B: [B,C]\times [A,B] \to [A,C]

The category [A,A][A,A] has a unit object – it would be churlish not to overload our notation yet further by calling it AA, corresponding to the fact that the free AA-model on one generator has a canonical co-AA-structure.

So we have a bicategory; the 0-cells are algebraic theories, the 1-cells are bimodels and the 2-cells are homomorphisms of bimodels. Consider a monad in this bicategory: an algebraic theory AA, an (A,A)(A,A)-bimodel MM, and homomorphisms η:AM\eta : A\to M, μ:M AMM\mu : M\otimes_A M \to M satisfying the usual rules. A module? of this monad is given by an AA-model BB together with an action M ABBM\otimes_A B \to B satisfying the usual rules. It should be clear that such modules are models of an algebraic theory, which we shall confusingly denote by MM. This theory is an extension of AA by unary operations (the elements of the underlying set of the underlying AA-model of the underlying (A,A)(A,A)-bimodel of the monad). The rules for composing them are given by μ\mu. They satisfy distributive laws over the operations of AA given by the co-AA-structure of MM.

We may overload η\eta to refer both to a homomorphism of bimodels and to a map of algebraic theories. The forgetful functor MModAModM Mod \to A Mod has for its left adjoint the functor M A?M\otimes_A ?, but it also has a right adjoint Hom A(M,?)Hom_A(M,?). So in this case the forgetful functor preserves colimits as well as limits. In fact all maps of theories whose associated forgetful functors have right adjoints must arise from such a monad in the bicategory of bimodels.

I would like some snappier terminology at this point. What should we call these monads in the bicategory of bimodels? If we use words like algebra or monad our rickety overloaded onomastic scaffolding starts to creak ominously. Put on your hard hats. We are in territory where to discriminate too meticulously between different views of the same thing is to invite fuddlement. And yet we have to hold in our heads that isomorphism is not equality, and that too cavalier an approach to identification can sometimes lead to error.

If AA were a ring, then I'd call MM an ‘AA-algebra’. Unfortunately, that term can also be used for an AA-model. Also, even in ring theory, that term is usually only used when AA is commutative. One might, following ‘bimodule’ (and ‘bimodel’) say ‘bialgebra’ in that case, but that also has another meaning. So let's give up in that direction.

But it seems OK to me to call it an ‘AA-monad’. —Toby

This fits with the fact that MM is an extension of AA by unary operations, so one should be reminded of monoids, maybe? —Gavin

Last revised on June 24, 2009 at 20:12:38. See the history of this page for a list of all contributions to it.