| Summary: | 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 nil‐Coxeter 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 towards the structure of a potential Floer homology theory of 3‐manifolds with codimension‐2 corners, we present a decomposition theorem for the Floer complex of a planar grid diagram, with respect to vertical and horizontal slicing.
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