  We develop a general framework for $c$-vectors of 2-Calabi--Yau categories,
which deals with cluster tilting subcategories that are not reachable from each
other and contain infinitely many indecomposable objects. It does not rely on
iterative sequences of mutations.
  We prove a categorical (infinite-rank) generalization of the
Nakanishi--Zelevinsky duality for $c$-vectors and establish two formulae for
the effective computation of $c$-vectors -- one in terms of indices and the
other in terms of dimension vectors for cluster tilted algebras.
  In this framework, we construct a correspondence between the $c$-vectors of
the cluster categories ${\mathscr{C}}(A_{\infty})$ of type $A_{\infty}$ due to
Igusa--Todorov and the roots of the Borel subalgebras of
${\mathfrak{sl}}_{\infty}$. Contrary to the finite dimensional case, the Borel
subalgebras of ${\mathfrak{sl}}_{\infty}$ are not conjugate to each other. On
the categorical side, the cluster tilting subcategories of
${\mathscr{C}}(A_{\infty})$ exhibit different homological properties. The
correspondence builds a bridge between the two classes of objects.
