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Homotopy Colimits of DG Categories and Fukaya Categories

2021/09/08 by Doğancan Karabaş, Sangjin Lee, Karabas, Dogancan +1
Mathematics · #18G35 #18N40 #53D37 #55P35 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.AT #math.CT #math.SG #msc:18G35 #msc:18N40 #msc:53D37 #msc:55P35

paper · pdf · doi:10.48550/arxiv.2109.03411

64 pages. Definition 2.14 is corrected in the second version. Definition 2.36, Definition 2.37, Theorem 2.73, and a mistake in the statement of Theorem 1.2 (Theorem 2.70) are corrected in the third version

openalex publication_date 2021/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a new cylinder object for semifree differential graded (dg) categories in the category of dg categories. Using this, we give a practical formula computing homotopy colimits of semifree dg categories. Combining it with the result of Ganatra, Pardon, and Shende, we get a formula computing wrapped Fukaya categories of Weinstein manifolds using their sectorial coverings. This formula has lots of applications including a practical computation of the wrapped Fukaya category of any cotangent bundle or plumbing space. In this paper, we compute wrapped Fukaya categories of cotangent bundles of lens spaces using their Heegaard decomposition. From the computation, we show that the endomorphism algebra of the cotangent fibre is a full invariant of the homotopy type of lens spaces.

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