{"arxiv_id":"2308.05089","incoming":{"papers":2,"sentences":5,"groups":[{"kind":"extends","label":"Builds on it","papers":1,"sentences":2,"quotes":[{"arxiv_id":"2312.14429","title":"Lagrangian cobordism and shadow distance in Tamarkin category","url":"/paper/2312.14429","section":"Guillermou doubling functor","sentence":"For a conical Lagrangian $Λ⊂T^*_τ> 0(M ×_t)$, we can define [equation] When $Λ/ _>0$ is compact, Guillermou [ref] defined a doubling functor from the Kashiwara–Schapira stack $_Λ(Λ)$ to a simple sheaf $F_dbl$, whose microsupport [equation] In the previous work of the third author [ref] , we generalized the theorem to the case where $Λ/ _>0$ is non-compact under a suitable assumption, that there exists a tubular neighborhood of the Lagrangian submanifold of positive radius with respect to some Riemannian metric adapted to the symplectic form.","intent_p":0.97,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses its method or tool","papers":1,"sentences":1,"quotes":[{"arxiv_id":"2308.05727","title":"Existence of generating families on Lagrangian cobordisms","url":"/paper/2308.05727","section":"Homotopy lifting of generating families","sentence":"In the similar construction for microlocal sheaves [ref] , following [ref] and [ref] , we separate the Legendrian fronts of $T_-ϵ(L)$ and $T_ϵ(L)$ by applying a contact isotopy that sends $T_-ϵ(L) ∪T_ϵ(L)$ by $T_-h(L) ∪T_h(L)$ for $r ≫0$ using a cut-off of the Reeb flow.","intent_p":0.88,"contests_p":null}]}],"unlabelled_sentences":2},"outgoing":{"papers":24,"sentences":62,"groups":[{"kind":"comparison","label":"Compares with","papers":2,"sentences":3,"quotes":[{"arxiv_id":"2108.10914","title":"Lagrangian cobordism functor in microlocal sheaf theory I","url":"/paper/2108.10914","section":"Computation of Lagrangian handle attachments","sentence":"Therefore, we will only give computations of Lagrangian handle attachments (the reader may compare with [ref] ).","intent_p":0.83,"contests_p":null},{"arxiv_id":"2012.06931","title":"Algebraic Weaves and Braid Varieties","url":"/paper/2012.06931","section":"Context of Lagrangian cobordism","sentence":"We remark that, though less general than the setting first paper [ref] , the setting in this paper is more intuitive and straightforward from the geometric perspective, and is the one considered in recent works on Legendrian weaves [ref] .","intent_p":0.92,"contests_p":null}]},{"kind":"supports_this_paper","label":"Cites as support","papers":1,"sentences":1,"quotes":[{"arxiv_id":"2008.10793","title":"Augmentations, Fillings, and Clusters","url":"/paper/2008.10793","section":"Construction of Legendrians with simple sheaf categories","sentence":"For instance, we now know a number of Legendrian links with different Lagrangian fillings such that the images of local systems on the different fillings have overlaps (for example, when the moduli space of rank 1 local systems on the fillings induced different cluster charts on the moduli space of microlocal rank 1 sheaves [ref] ).","intent_p":0.97,"contests_p":null}]},{"kind":"background","label":"Cites as background","papers":8,"sentences":18,"quotes":[{"arxiv_id":"1212.5818","title":"Quantization of conic Lagrangian submanifolds of cotangent bundles","url":"/paper/1212.5818","section":"Sheaf Quantization of Lagrangian Cobordisms","sentence":"Following the convention of [ref] , sheaf quantization is a functor from Lagrangian submanifolds $L ⊂T^*M$ with local systems to sheaves with singular support on the Legendrian lift, i.e. [equation] For conical Lagrangian cobordisms, we do not have a sheaf quantization functor in general.","intent_p":0.91,"contests_p":null},{"arxiv_id":"1704.04291","title":"Brane structures in microlocal sheaf theory","url":"/paper/1704.04291","section":"Sheaf Quantization of Lagrangian Cobordisms","sentence":"Let $T_u: J^1(M ×ℝ_>0) →J^1(M ×ℝ_>0)$ be the Reeb flow.Guillermou and Jin-Treumann [ref] considered the sheaf quantization of closed exact Lagrangians or exact Lagrangians with positive conical ends.","intent_p":0.85,"contests_p":null},{"arxiv_id":"2007.10154","title":"Sheaf quantization in Weinstein symplectic manifolds","url":"/paper/2007.10154","section":"Doubling Functor for Noncompact Legendrians","sentence":"The doubling construction goes back to Guillermou [ref] (and Jin-Treumann [ref] ), who defined an unconditional doubling functor for a compact exact Lagrangian (or an exact Lagrangian with positive conical ends) [equation] and is also formulated in a different way in Nadler-Shende [ref] .","intent_p":0.85,"contests_p":null},{"arxiv_id":"1604.00114","title":"Wrapped microlocal sheaves on pairs of pants","url":"/paper/1604.00114","section":"Structure of Kashiwara-Schapira stack","sentence":"We review the definition and properties of microlocalization and the sheaf of categories $μSh$, which has been introduced and studied in [ref] *Section 6, [ref] *Section 6 or [ref] *Section 3.4.","intent_p":1.0,"contests_p":null},{"arxiv_id":"2108.10914","title":"Lagrangian cobordism functor in microlocal sheaf theory I","url":"/paper/2108.10914","section":"Computation of Lagrangian handle attachments","sentence":"Lagrangian cobordisms induced by Legendrian isotopies have been studied by Guillermou-Kashiwara-Schapira [ref] .","intent_p":0.92,"contests_p":null}]}],"unlabelled_sentences":40},"claims":[],"thresholds":{"intent_p":0.8,"supports_p":0.95,"contests_p":0.7},"method":"Citing sentences come from each paper's LaTeX source; every \\cite marks the sentence and the works it cites. Pith's classifier typed each sentence. A label is shown at intent probability 0.8 or more (0.95 for support), a contest at 0.7 or more. Measured accuracy against two blind model families is at https://pith.science/accuracy. The quoted sentences are the citing authors' own.","held_up":null,"_links":{"html":"https://pith.science/paper/2308.05089"}}