A function or differential form on a manifold with boundary to a is said to have sitting instants at the boundary if there is an open neighbourhood $\partial X \subset U \subset X$ of the boundary such that the restriction $f|_U$ of $f$ to that neighbourhood is locally constant? in the direction perpendicular to the boundary.
Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Def. 4.2.1 in_Cech Cocycles for Differential characteristic Classes, Adv. in Theor. and Math. Phys., Volume 16 Issue 1 (2012), pages 149-250
Vincent Braunack-Mayer, Def. 3.1.3 Rational parameterized stable homotopy theory, Zurich 2018 (pdf)
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