Let ${\mathrm{\Lambda }}_{\pm }$ be Legendrian submanifolds in the cosphere bundle $T^{\ast ,\infty }M$. Given a Lagrangian cobordism $L$ of Legendrians from ${\mathrm{\Lambda }}_{-}$ to ${\mathrm{\Lambda }}_{+}$, we construct a functor ${\mathrm{\Phi }}_L^{*}:{\mathrm{Sh}}_{{\mathrm{\Lambda }}_{+}}^c(M)\to {\mathrm{Sh}}_{{\mathrm{\Lambda }}_{-}}^c(M){\otimes }_{C_{-*}({\mathrm{\Omega }}_{*}{\mathrm{\Lambda }}_{-})}{C}_{-*}({\mathrm{\Omega }}_{*}L)$ between sheaf categories of compact objects with singular support on ${\mathrm{\Lambda }}_{\pm }$ and its right adjoint on sheaf categories of proper objects, using Nadler–Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high-dimensional Legendrian submanifolds.

中文翻译：

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微局域层理论中的拉格朗日协边函子 I

让${\mathrm{\Lambda }}_{\pm }$是余球丛中的勒让德子流形${\mathrm{时间}}^{*,\mathrm{无穷大}}\mathrm{中号}$。给定拉格朗日配边$L$传奇人物来自${\mathrm{\Lambda }}_{-}$到${\mathrm{\Lambda }}_{+}$，我们构造一个函子${\mathrm{\Phi }}_L^{*}:{什}_{{\mathrm{\Lambda }}_{+}}^C（\mathrm{中号}）\to {什}_{{\mathrm{\Lambda }}_{-}}^C（\mathrm{中号}）{\otimes }_{C_{-*}（{\mathrm{\Omega }}_{*}{\mathrm{\Lambda }}_{-}）}{C}_{-*}（{\mathrm{\Omega }}_{*}L）$在具有单一支撑的紧凑物体的束类别之间${\mathrm{\Lambda }}_{\pm }$以及它在真对象的层范畴上的右伴随，使用 Nadler–Shende 的工作。这给出了类似于勒格朗日接触同调上的拉格朗日配边图及其单位增广范畴上的右伴随的层理论描述。我们还推导了一些长精确序列和高维勒格朗德子流形之间的拉格朗日配边的新障碍。