  We begin the development of a categorical perspective on the theory of
generalized interval systems (GIS's). Morphisms of GIS's allow the analyst to
move between multiple interval systems and connect transformational networks.
We expand the analytical reach of the Sub Dual Group Theorem of Fiore--Noll
(2011) and the generalized contextual group of Fiore--Satyendra (2005) by
combining them with a theory of GIS morphisms. Concrete examples include an
analysis of Schoenberg, String Quartet in D minor, op. 7, and simply transitive
covers of the octatonic set. This work also lays the foundation for a
transformational study of Lawvere--Tierney upgrades in the topos of triads of
Noll (2005).
