  The paper studies the Karoubi envelope of a one-dimensional topological
theory with defects and inner endpoints, defined over a field. It turns out
that the Karoubi envelope is determined by a symmetric Frobenius algebra K
associated to the theory. The Karoubi envelope is then equivalent to the
quotient of the Frobenius-Brauer category of K modulo the ideal of negligible
morphisms. Symmetric Frobenius algebras, such as K, describe two-dimensional
TQFTs for the category of thin flat surfaces, and elements of the algebra can
be turned into defects on the side boundaries of these surfaces. We also
explain how to couple K to the universal construction restricted to closed
surfaces to define a topological theory of open-closed two-dimensional
cobordisms which is usually not an open-closed 2D TQFT.
