  Laplante-Anfossi associated to each rooted plane tree a polytope called an
operahedron. He also defined a partial order on the vertex set of an
operahedron and asked if the resulting poset is a lattice. We answer this
question in the affirmative, motivating us to name Laplante-Anfossi's posets
operahedron lattices. The operahedron lattice of a chain with $n+1$ vertices is
isomorphic to the $n$-th Tamari lattice, while the operahedron lattice of a
claw with $n+1$ vertices is isomorphic to $\mathrm{Weak}(\mathfrak S_n)$, the
weak order on the symmetric group $\mathfrak S_n$. We characterize
semidistributive operahedron lattices and trim operahedron lattices. Let
$\Delta_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ be the principal order
ideal of $\mathrm{Weak}(\mathfrak S_n)$ generated by the permutation
${w_\circ(k,n)=k(k-1)\cdots 1(k+1)(k+2)\cdots n}$. Our final result states that
the operahedron lattice of a broom with $n+1$ vertices and $k$ leaves is
isomorphic to the subposet of $\mathrm{Weak}(\mathfrak S_n)$ consisting of the
preimages of $\Delta_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ under West's
stack-sorting map; as a consequence, we deduce that this subposet is a
semidistributive lattice.
