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Heuristically, it is known that Courant algebroids should "integrate" to symplectic 2-groupoids, but very little of this correspondence has been developed in a precise way. I will describe in detail the case of a linear 2-groupoid equipped with a constant symplectic form, and I will explain how these "constant symplectic 2-groupoids" correspond to a certain class of Courant algebroids. The study of constant symplectic 2-groupoids is intended to be a first step toward a more general study of symplectic 2-groupoids, in analogy to how a student usually first learns about symplectic vector spaces before moving on to symplectic manifolds. Symplectic 2-groupoids are closely related to the shifted symplectic structures studied by Pantev, et al, although the definition is more "strict" in certain ways. As part of the talk, I will give some context to explain why the additional strictness is appropriate for the problem of integrating Courant algebroids.