even, and if it is in ${\cal S}_{\cal B}^{\ast t}$, then they are odd. We will now define a category ${\cal L}^{\cal A}_\omega$, isomorphic to ${\cal
S}_{\cal A}^{\ast t}$. This category, made out of the material of the monoid ${\cal L}_\omega$, interests us only as a stepping stone towards the isomorphic geometric categories *Frz* and *Frobse*, which we will consider in a moment.

The objects of the category ${\cal L}^{\cal A}_\omega$ are again the natural numbers. An arrow of this category will be obtained from an arrow ${\langle [f], n, m\rangle}$ of ${\cal S}_{\cal
A}^{\ast t}$ by replacing the class $[f]$ by the corresponding element $e$ of ${\cal L}_\omega$, and $n$ and $m$ by respectively $n/2$ and $m/2$; the type of ${\langle e,n/2,m/2\rangle}$ in ${\cal L}^{\cal A}_\omega$ is ${n/2\str m/2}$. We divide the numbers in the types by two to obtain types that will correspond to the types in *Frob* (see the preceding section).

In this way, with every element of ${\cal L}_\omega$ we associate in ${\cal L}^{\cal A}_\omega$ a denumerable infinity of types. The generator $a^\alpha_{2n+1}$ of ${\cal L}_\omega$ (see [@DP03], Section on *Normal forms in* ${\cal L}_\omega$) will have as associated types $n\pl l\pl 1\str n\pl l$, for 