  We present a state sum construction of two-dimensional extended Topological
Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes
the state sum of Fukuma--Hosono--Kawai from triangulations of conventional
two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth
compact oriented 2-manifolds with corners that have a particular global
structure. This construction reveals the topological interpretation of the
associative algebra on which the state sum is based, as the vector space that
the TQFT assigns to the unit interval. Extending the notion of a
two-dimensional TQFT from cobordisms to suitable manifolds with corners
therefore makes the relationship between the global description of the TQFT in
terms of a functor into the category of vector spaces and the local description
in terms of a state sum fully transparent. We also illustrate the state sum
construction of an open-closed TQFT with a finite set of D-branes using the
example of the groupoid algebra of a finite groupoid.
