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Let X X be a compact real analytic manifold, and let T ∗ X T^*X be its cotangent bundle. Let S h (X) Sh (X) be the triangulated dg category of bounded, constructible complexes of sheaves on X X. In this paper, we develop a Fukaya A ∞ A_ -category F u k (T ∗ X) Fuk (T^*X) whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write T w F u k (T ∗ X) Tw Fuk (T^*X) for the A ∞ A_ -triangulated envelope of F u k (T ∗ X) Fuk (T^*X) consisting of twisted complexes of Lagrangian branes. Our main result is that S h (X) Sh (X) quasi-embeds into T w F u k (T ∗ X
Nadler et al. (Wed,) studied this question.