The algebraic structure consisting of oriented closed 1-manifolds, 2-dimensional bordisms between them, and 3-dimensional bordisms between those, forms a 2-category B. A representation of B is called an "oriented 1-2-3 topological quantum field theory". Such a representation can be thought of as a representation of the structure formed by low-dimensional manifolds: it assigns invariants to 3-manifolds, vector spaces to 2-manifolds, and linear categories to 1-manifolds in a functorial way. I will outline a generators-and relations description of B obtained from parameterized Morse theory, which I will use to give an explicit description of the representations of B. Joint work with Jamie Vicary, Christopher Douglas and Chris Schommer-Pries.