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\begin{document}

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\section*{bimodel}

Let $A$ and $B$ be [[algebraic theory|algebraic theories]]. The category $[A,B]$ of \textbf{$(A,B)$-bimodels} and their homomorphisms is the category of $A$-models and homomorphisms in $B Mod^{op}$. An alternative description is that is a co-$A$-model in $B Mod$. Each such bimodel $M$ determines and is determined by a pair of [[adjoint functor]]s

\begin{displaymath}
Hom_B(M,?): B Mod \to A Mod
\end{displaymath}
\begin{displaymath}
M\otimes_A ?: A Mod \to B Mod
\end{displaymath}
Composition of such adjoint pairs yields a functor

\begin{displaymath}
\otimes_B: [B,C]\times [A,B] \to [A,C]
\end{displaymath}
The category $[A,A]$ has a unit object -- it would be churlish not to overload our notation yet further by calling it $A$, corresponding to the fact that the free $A$-model on one generator has a canonical co-$A$-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 $A$, an $(A,A)$-bimodel $M$, and homomorphisms $\eta : A\to M$, $\mu : M\otimes_A M \to M$ satisfying the usual rules. A [[algebra of a monad|module]] of this monad is given by an $A$-model $B$ together with an action $M\otimes_A B \to B$ satisfying the usual rules. It should be clear that such modules are models of an algebraic theory, which we shall confusingly denote by $M$. This theory is an extension of $A$ by unary operations (the elements of the underlying set of the underlying $A$-model of the underlying $(A,A)$-bimodel of the monad). The rules for composing them are given by $\mu$. They satisfy [[distributive law]]s over the operations of $A$ given by the co-$A$-structure of $M$.

We may overload $\eta$ to refer both to a homomorphism of bimodels and to a map of algebraic theories. The forgetful functor $M Mod \to A Mod$ has for its left adjoint the functor $M\otimes_A ?$, but it also has a right adjoint $Hom_A(M,?)$. So in this case the forgetful functor preserves colimits as well as limits. In fact all maps of theories whose associated forgetful functors have [[adjoint functor|right adjoint]]s 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 \emph{algebra} or \emph{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 $A$ were a ring, then I'd call $M$ an `$A$-algebra'. Unfortunately, that term can also be used for an $A$-model. Also, even in ring theory, that term is usually only used when $A$ 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 `$A$-monad'. ---Toby

This fits with the fact that $M$ is an extension of $A$ by unary operations, so one should be reminded of monoids, maybe? ---Gavin



\end{document}