Under an assumption on the existence of
$p$
-adic Galois representations, we carry out Taylor–Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated with
$\operatorname{GL}_{n}$
over a number field. We use our construction, and some new results in non-commutative algebra, to show that standard conjectures on completed homology imply ‘big
$R=\text{big}~\mathbb{T}$
’ theorems in situations where one cannot hope to appeal to the Zariski density of classical points (in contrast to all previous results of this kind). In the case where
$n=2$
and
$p$
splits completely in the number field, we relate our construction to the
$p$
-adic local Langlands correspondence for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
.