  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.
