A Galois connection between subalgebras and tensor subcategories

Let $\mathcal{B}$ be a braided finite tensor category and let $A$ be a simple commutative algebra in $\mathcal{B}$. We construct an order-reversing Galois connection between subalgebras of $A$ and tensor subcategories of ${\mathcal{B}}_A$. Let ${\mathcal{B}}^{\mathrm{\prime }}$ denote the Müger center and set $A^{\mathrm{\prime }}:=A\cap {\mathcal{B}}^{\mathrm{\prime }}$. The closure operators are $B\mapsto ⟨B,A^{\mathrm{\prime }}{⟩}_{\mathrm{a}\mathrm{l}\mathrm{g}}$ and $\mathcal{E}\mapsto ⟨\mathcal{E},{\mathcal{B}}_A^{\mathrm{l}\mathrm{o}\mathrm{c}}{⟩}_{\otimes }$. Thus the closed subalgebras are those containing $A^{\mathrm{\prime }}$, while the closed tensor subcategories are those containing ${\mathcal{B}}_A^{\mathrm{l}\mathrm{o}\mathrm{c}}$; equivalently, the fixed-point intervals $[A^{\mathrm{\prime }},A{]}_{\mathrm{a}\mathrm{l}\mathrm{g}}$ and $[{\mathcal{B}}_A^{\mathrm{l}\mathrm{o}\mathrm{c}},{\mathcal{B}}_A{]}_{\otimes }$ are anti-isomorphic as lattices. For a finite tensor category $\mathcal{C}$, the canonical algebra in $\mathcal{Z}(\mathcal{C})$ yields an anti-isomorphism between its subalgebras and tensor subcategories of $\mathcal{C}$. When $\mathcal{B}$ is nondegenerate, Frobenius extensions correspond to unimodular tensor subcategories. For a Hopf algebra in $\mathcal{B}$, the correspondence specializes to an order-preserving bijection between Hopf ideals and normal left coideal subalgebras.