$A_{\infty }$-category of Lagrangian cobordisms in the symplectization of $P\times \mathbb{R}$

Abstract

We define a unital $A_{\infty }$-category $Fuk(\mathbb{R}\times Y)$ whose objects are exact Lagrangian cobordisms in the symplectization of $Y=P\times \mathbb{R}$, with negative cylindrical ends over Legendrians equipped with augmentations. The morphism spaces ${\mathrm{hom}}_{Fuk(\mathbb{R}\times Y)}({\Sigma }_0,{\Sigma }_1)$ are given in terms of Floer complexes ${\mathrm{Cth}}_{+}({\Sigma }_0,{\Sigma }_1)$ which are versions of the Rabinowitz Floer complex defined by Symplectic Field Theory (SFT) techniques.