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

$c$-vectors of 2-Calabi--Yau categories and Borel subalgebras of ${\mathfrak{sl}}_{\infty}$