  We study a correspondence associating to each subshift $\mathcal X$ of
$A^{\mathbb Z}$ a subcategory of the Karoubi envelope of the free profinite
semigroup generated by $A$. The objects of this category are the idempotents in
the mirage of $\mathcal X$, that is, in the set of pseudowords whose finite
factors are blocks of $\mathcal X$. The natural equivalence class of the
category is shown to be invariant under flow equivalence. As a corollary of our
proof, we deduce the flow invariance of the profinite group that Almeida
associated to each irreducible subshift. We also show, in a functorial manner,
that the isomorphism class of the category is invariant under conjugacy.
Finally, we see that the zeta function of $\mathcal X$ is naturally encoded in
the category. These results hold, with obvious translations, for relatively
free profinite semigroups over many pseudovarieties, including all of the form
$\overline{\mathsf H}$, with $\mathsf H$ a pseudovariety of groups.
