To a Legendrian knot, one can associate an
category, the augmentation category. An exact Lagrangian cobordism between two
Legendrian knots gives a functor of the augmentation categories of the two
knots. We study this functor and establish a long exact sequence relating the
corresponding cohomology of morphisms of the two ends. As applications, we prove
that the functor between augmentation categories is injective on the level of
equivalence classes of objects and find new obstructions to the existence of exact
Lagrangian cobordisms in terms of linearized contact homology and ruling
polynomials.
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