Charles, Julien Marché and Raphael Zentner, for their interest, and helpful conversations.

Floer Field Theory {#sec:fft}
==================

Outline {#sec:outline}
-------

According to Wehrheim and Woodward [@WWfft], a *Floer field theory* is a functor from a category of (2+1)-cobordisms, to a variation of Weinstein’s symplectic category, whose objects are symplectic manifolds, and morphisms (equivalence classes of) sequences of Lagrangian correspondences (see Definition \[def:lagcorresp\]).

Such a functor being given, one can associate a topological invariant to a closed oriented 3-manifold $Y$ in the following way: after removing two 3-balls to $Y$, one obtains a cobordism from the 2-sphere to itself, the functor applied to this cobordism gives rise to a sequence of Lagrangian correspondences having the same source and target symplectic manifolds, and to such a sequence (with extra technical assumptions) one can associate a homology group called quilted Floer homology, which is a generalization of Lagrangian Floer homology.

We will see that HSI groups fit into this framework, as already suggested in Manolescu and Woodward’s proof of their stabilisation invariance. Yet, some slight modifications will be necessary, mainly for three reasons:

-   The moduli spaces are associated to surfaces with one boundary component: a disc will have to be removed to 