  We define a new homology theory we call symbol homology by using decorated
moduli spaces of Whitney polygons. By decorating different types of moduli
spaces we obtain different flavors of this homology theory together with
morphisms between them. Each of these flavors encodes the properties of a
different type of Heegaard Floer homology. The morphisms between the symbol
homologies enable us to push properties from one Floer theory to a different
one. Furthermore, we obtain a new presentation of Heegaard Floer theory in
which maps correspond to multiplication from the right with suitable elements
of our symbol homology. Finally, we present the construction of cobordism maps
in knot Floer theories and apply the tools from symbol homology to give an
invariance proof.
