This is less of a strictly mathematical question and more of a reference request.

In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston (LOT) associate a dg algebra $\mathcal{A(F)}$ to a parametrized surface $F$. To a 3-manifold bounding a parametrized surface $F$, they associate (among other things) a "Type D structure" over $\mathcal{A}(F)$. Type D structures turn out to be (modulo concerns about finiteness) the same as twisted complexes over $\mathcal{A}(F)$, as defined by Bondal and Kapranov in "Framed Triangulated Categories" (this was pointed out by LOT in Remark 2.25 of the above paper).

In this setting, to a cobordism between two parametrized surfaces $F$ and $F'$, LOT associate a "DA bimodule," which is more or less a bimodule over $\mathcal{A}(F)$ and $\mathcal{A}(F')$ which acts like a type $D$ structure over $F'$ and an $A_{\infty}$ module over $F$.

A DA bimodule over $\mathcal{A}(F)$ and $\mathcal{A}(F')$ defines a functor between the corresponding categories of twisted complexes or Type D structures, using the box tensor product operation $\boxtimes$ which is also defined by LOT.

My question is: in the setting of twisted complexes, which existed before Heegaard Floer homology, was there a preexisting definition corresponding to LOT's DA bimodules? How about to the box tensor product operation?

Alternatively, if these definitions are new to LOT's work, do they contribute anything new to older questions about twisted complexes (not having to do with Heegaard Floer homology)?


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