  We introduce the notion of a Lie semiheap as a smooth manifold equipped with
a para-associative ternary product. For a particular class of Lie semiheaps we
establish the existence of left-invariant vector fields. Furthermore, we show
how such manifolds are related to Lie groups and establish the analogue of
principal bundles in this ternary setting. In particular, we generalise the
well-known `heapification' functor to the ambience of Lie groups and principal
bundles.
