  As a generalization of acyclic 2-Calabi-Yau categories, we consider
2-Calabi-Yau categories with a directed cluster-tilting subcategory; we study
their cluster-tilting subcategories and the cluster combinatorics that they
encode. We show that such categories have a cluster structure.
  Triangulated 2-Calabi-Yau categories with a directed cluster-tilting
subcategory are closely related to representations of certain semi-hereditary
categories, more specifically to representations of thread quivers. Thread
quivers are a tool to classify and study certain semi-hereditary categories
using both quivers and linearly ordered sets (threads).
  We study the case where the thread quiver consists of a single thread (so
that representations of this thread quiver correspond to representations of
some linearly ordered set), and show that, similar to the case of a Dynkin
quiver of type $A$, the cluster-tilting subcategories can be understood via
triangulations of an associated cyclically ordered set.
  In this way, we gain insight into the structure of the cluster-tilting
subcategories of 2-Calabi-Yau categories with a directed cluster-tilting
subcategory. As an application, we show that every 2-Calabi-Yau category which
admits a directed cluster-tilting subcategory with countably many isomorphism
classes of indecomposable objects has a cluster-tilting subcategory
$\mathcal{V}$ with the following property: any rigid object in the cluster
category can be reached from $\mathcal{V}$ by finitely many mutations. This
implies that there is a cluster map which is defined on all rigid objects, and
thus that there is a cluster algebra whose cluster variables are exactly given
by the rigid indecomposable objects.
