a closed surface, and similarly, a tube connecting two such discs will have to be removed to a cobordism. Since the Lagrangian correspondence obtained will depend on the choice of this tube (as in Example \[exemchgtchemin\]), these should be incorporated in the cobordism category. Rather than being seen as an inconvenient, this phenomena might be used to define invariants for knots and sutured manifolds. Moreover, Lagrangian corresponces will also depend on a parametrization of the tube (see Example \[exemreparam\]), of which the category should also keep track.

-   The twisting homology class in $H_1(Y;{{\mathbb{Z}}_{2}})$ should also be incorporated in the cobordism. Although this class corresponds to a second Stiefel-Whitney class, we prefer working with homology classes to avoid using relative cohomology classes.

-   The target symplectic category will also have to be more complicated in order to be able to define quilted Floer homology, for the reasons described in section \[sec:outlineproofs\] : symplectic manifolds will come equipped with a hypersurface, and two 2-forms, which should satisfy extra technical assumptions as in [@MW Assumption 2.5].

Quilts, the symplectic category, and Floer homology {#sec:quiltsetc}
---------------------------------------------------

\[def:lagcorresp\] A *Lagrangian correspondence* between two symplectic manifolds $M$ and $M'$ is a Lagrangian submanifold $L\subset M^- 