A symplectic Lie groupoid is a Lie groupoid with a 
multiplicative symplectic form. We take the perspective that such an 
object is symplectic manifold with an extra categorical structure. 
Applying the machinery of Floer theory, the extra structure is expected 
to yield a monoidal structure on the Fukaya category, and new operations 
on the closed string invariants. I will take an examples-based approach 
to working out what these structures are, focusing on cases where the 
Floer theory is tractable, such as the cotangent bundle of a compact 
manifold.