  In this note, we classify topological solitons of $n$-brane fields, which are
nonlocal fields that describe $n$-dimensional extended objects. We consider a
class of $n$-brane fields that formally define a homomorphism from the $n$-fold
loop space $\Omega^n X_D$ of spacetime $X_D$ to a space $\mathcal{E}_n$.
Examples of such $n$-brane fields are Wilson operators in $n$-form gauge
theories. The solitons are singularities of the $n$-brane field, and we
classify them using the homotopy theory of ${\mathbb{E}_n}$-algebras. We find
that the classification of codimension ${k+1}$ topological solitons with
${k\geq n}$ can be understood using homotopy groups of $\mathcal{E}_n$. In
particular, they are classified by ${\pi_{k-n}(\mathcal{E}_n)}$ when ${n>1}$
and by ${\pi_{k-n}(\mathcal{E}_n)}$ modulo a ${\pi_{1-n}(\mathcal{E}_n)}$
action when ${n=0}$ or ${1}$. However, for ${n>2}$, their classification goes
beyond the homotopy groups of $\mathcal{E}_n$ when ${k< n}$, which we explore
through examples. We compare this classification to $n$-form $\mathcal{E}_n$
gauge theory. We then apply this classification and consider an ${n}$-form
symmetry described by the abelian group ${G^{(n)}}$ that is spontaneously
broken to ${H^{(n)}\subset G^{(n)}}$, for which the order parameter
characterizing this symmetry breaking pattern is an ${n}$-brane field with
target space ${\mathcal{E}_n = G^{(n)}/H^{(n)}}$. We discuss this
classification in the context of many examples, both with and without 't Hooft
anomalies.
