  The goal of this article is to clarify the relationship between the topos of
triads and the neo-Riemannian PLR-group. To do this, we first develop some
theory of generalized interval systems: 1) we prove the well known fact that
every pair of dual groups is isomorphic to the left and right regular
representations of some group (Cayley's Theorem), 2) given a simply transitive
group action, we show how to construct the dual group, and 3) given two dual
groups, we show how to easily construct sub dual groups. Examples of this
construction of sub dual groups include Cohn's hexatonic systems, as well as
the octatonic systems. We then enumerate all Z_{12}-subsets which are invariant
under the triadic monoid and admit a simply transitive PLR-subgroup action on
their maximal triadic covers. As a corollary, we realize all four hexatonic
systems and all three octatonic systems as Lawvere--Tierney upgrades of
consonant triads.
