Cyclic structures and broken cycles

Abstract

We introduce a new way to encode semicyclic structures using a stack of broken cycles. (We also prove an analogue for paracyclic structures.) This was motivated not only by higher algebra but also by Fukaya-categorical considerations. We also openly speculate about some Fukaya-categorical implications. For example, this stack sees moduli of stopped Liouville disks, and hence yields another platform for gluing together Fukaya categories. We also see that Lagrangian cobordisms with multiple ends may not only serve to detect $K_0$ groups of Fukaya categories, but higher $K$-theory groups as well. Along the way, we include brief expositions of (i) basic techniques in infinity-categories and (ii) the translation between exit path categories and constructible sheaves. These may be of independent interest.

References in corpus (8)

• Covariantly functorial wrapped Floer theory on Liouville sectors
• Higher Segal spaces I
• Triangulated surfaces in triangulated categories
• Arboreal Singularities in Weinstein Skeleta
• A stable infinity-category of Lagrangian cobordisms
• The Fukaya category pairs with Lagrangian cobordisms exactly
• Cyclic symmetries of A_n-quiver representations
• Associative algebras and broken lines

Cited by in corpus (8)

• The symplectic geometry of higher Auslander algebras: Symmetric products of disks
• Orlov and Viterbo functors in partially wrapped Fukaya categories
• Smooth constructions of homotopy-coherent actions
• The stratified homotopy type of the reductive Borel-Serre compactification
• Continuous and coherent actions on wrapped Fukaya categories
• Spherical monadic adjunctions of stable infinity categories
• Lagrangian Cobordisms in Liouville manifolds
• The infinity-category of stabilized Liouville sectors