This article develops a model of narrative reference by showing that the organization of meaning in narrative is congruent with the structure of space and time. In making this connection, the article's principal methodological tool is mathematical catastrophe theory. Specifically, it is shown that narratives like Hesiod's Homeric Hymn to Demeter can be construed as hypersurfaces that permit a number of critical transformations between stable regions of thematic value. It is argued that this approach is superior to cognitivist theories of reference because it avoids the vicious circle of displacing the referential function of narrative on to psychic schemas that have themselves a narrative structure.