We study augmentations of a Legendrian surface
in the
–jet space,
, of a surface
. We
introduce two types of algebraic/combinatorial structures related to the front projection
of
that we call chain homotopy diagrams (CHDs) and Morse complex
–families
(MC2Fs), and show that the existence of a
–graded CHD or a
–graded MC2F is equivalent
to the existence of a
–graded
augmentation of the Legendrian contact homology DGA to
. A
CHD is an assignment of chain complexes, chain maps, and homotopy operators to
the
–,
–, and
–cells
of a compatible polygonal decomposition of the base projection of
with restrictions arising from the front projection of
.
An MC2F consists of a collection of formal handleslide sets and chain
complexes, subject to axioms based on the behavior of Morse complexes in
–parameter
families. We prove that if a Legendrian surface has a tame-at-infinity generating family, then it has
a
–graded MC2F
and hence a
–graded
augmentation. In addition, continuation maps and a monodromy representation of
are associated to augmentations, and then used to provide more refined
obstructions to the existence of generating families that (i) are linear at
infinity or (ii) have trivial bundle domain. We apply our methods in several
examples.
