derived smooth geometry
The notion of -plectic form is a generalization of the notion of symplectic form to differential forms of more than two arguments. This is considered in higher symplectic geometry, specifically: in n-plectic geometry/multisymplectic geometry.
An 1-plectic form is equivalently a symplectic form.
If the last condition is dropped, then by analogy with presymplectic forms one may speak of a pre--plectic form. Of course this is just a closed -form, but as in the presymplectic case, the plectic-terminology indicates that one wants to regard it as input datum for higher geometric quantization.
This definition has an evident generalization to the case where also is allowed to be a “higher” generalization of a smooth manifold, namely a smooth ∞-groupoid or L-∞ algebroid. See higher symplectic geometry for more on this case.
For instance definition 2.1 in