  Bordered Floer homology associates to a parametrized oriented surface a
certain differential graded algebra. We study the properties of this algebra
under splittings of the surface. To the circle we associate a differential
graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with
connected boundary an algebra-module over this 2-algebra, such that a natural
gluing property is satisfied. Moreover, with a view toward the structure of a
potential Floer homology theory of 3-manifolds with codimension-two corners, we
present a decomposition theorem for the Floer complex of a planar grid diagram,
with respect to vertical and horizontal slicing.
