  We define new algebras, local bimodules, and bimodule maps in the spirit of
Ozsvath-Szabo's bordered knot Floer homology. We equip them with the structure
of 2-representations of the categorified negative half U^- of U_q(gl(1|1)),
1-morphisms of such, and 2-morphisms respectively, and show that they
categorify representations of U_q(gl(1|1)^-) and maps between them. Unlike with
Ozsvath-Szabo's algebras, the algebras considered here can be built from a
higher tensor product operation recently introduced by Rouquier and the author.
  Our bimodules are all motivated by holomorphic disk counts in Heegaard
diagrams; for positive and negative crossings, the bimodules can also be
expressed as mapping cones involving a singular-crossing bimodule and the
identity bimodule. In fact, they arise from an action of the monoidal category
of Soergel bimodules via Rouquier complexes in the usual way, the first time
(to the author's knowledge) such an expression has been obtained for braiding
bimodules in Heegaard Floer homology.
  Furthermore, the singular crossing bimodule naturally factors into two
bimodules for trivalent vertices; such bimodules have not appeared in previous
bordered-Floer approaches to knot Floer homology. The action of the Soergel
category comes from an action of categorified quantum gl(2) on the
2-representation 2-category of U^- in line with the ideas of skew Howe duality,
where the trivalent vertex bimodules are associated to 1-morphisms E, F in
categorified quantum gl(2).
