A Topological Model for the Fukaya Categories of Plumbings
نویسندگان
چکیده
We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A∞ structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds Q1 and Q2 of the same dimension with embeddings of a submanifold B with isomorphic normal bundles, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A∞ category obtained by considering exact Lagrangian embeddings of Q1 and Q2 which intersect cleanly along B.
منابع مشابه
Fukaya Categories as Categorical Morse Homology
The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to t...
متن کاملTopological Conformal Field Theories and Calabi-yau Categories
This paper concerns open, closed, and open-closed topological conformal field theories. We show that the category of open topological conformal field theories, at all genera, is homotopy equivalent to a category of Calabi-Yau A∞ categories. For each such, we show that there is a universal closed TCFT, which is the initial element in the category of compatible open-closed TCFTs. The homology of ...
متن کاملCategorically-algebraic topology and its applications
This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...
متن کاملFukaya categories and deformations
Soon after their first appearance [7], Fukaya categories were brought to the attention of a wider audience through the homological mirror conjecture [14]. Since then Fukaya and his collaborators have undertaken the vast project of laying down the foundations, and as a result a fully general definition is available [9, 6]. The task that symplectic geometers are now facing is to make these catego...
متن کاملFunctorial semantics of topological theories
Following the categorical approach to universal algebra through algebraic theories, proposed by F.~W.~Lawvere in his PhD thesis, this paper aims at introducing a similar setting for general topology. The cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. We introduce t...
متن کامل