Cornered Heegaard Floer homology

Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 corners, and prove that the bordered Floer homology of a 3-manifold with boundary, split into two pieces with corners, can be recovered as a tensor product of the cornered invariants of the pieces.