  We prove that Lagrangian cocores and Lagrangian linking disks of a stopped
Weinstein manifold generate the Lagrangian cobordism infinity-category.
  As a geometric consequence, we see that any brane (after stabilization)
admits a Lagrangian cobordism to a disjoint union of some standard collection
of branes (cocores, linking disks, and a zero object). For example, when our
stopped Weinstein manifold is a point stopped by itself, we find that any exact
brane in Euclidean space admits a Lagrangian cobordism to a disjoint union of
cotangent fibers and a zero object. (This is a stronger statement than one
could obtain from purely Fukaya-categorical generation results.) Our methods
are constructive. For example, when our Weinstein manifold is a point, after
stabilization we can resolve the conormal to a compact manifold A of R^n by a
sequence of cotangent fibers; the resulting filtration realizes, after passage
to the wrapped Fukaya category, the Morse cochain complex of A associated to
(and hence filtered by) a generic ``distance to a point'' function; the
associated gradeds are the reduced homologies of the Morse attaching spheres.
  There is also an algebraic consequence. Lagrangian cobordism theory is
conjectured (in analogue to classical cobordism theory) to be linear over a
ring spectrum L controlling Lagrangian cobordisms between cotangent fibers in
Euclidean spaces. Our main theorem gives strong evidence for this conjecture:
The infinity-category of Lagrangians and their cobordisms in R^infinity is
equivalent to a full subcategory of modules over L.
  We conclude by proving a \pi_0-level theorem that gives further evidence of
the above conjecture: We exhibit a \pi_0-level symmetric monoidal structure
compatible with the linear structure of L-modules.
