Let $\mathcal{F}=(X,f,B)$ and $\mathcal{F}^{\prime}=(X^{\prime},f^{\prime},B^{\prime})$ be a pair of topological fibrations with discriminant loci $\Delta$ and $\Delta^{\prime}$ respectively. We define the following notions of conjugacy between $\mathcal{F}$ and $\mathcal{F}^{\prime}$:

• (i)

  We say that $\mathcal{F}$ is conjugate to $\mathcal{F}^{\prime}$ if there exist a homeomorphism $\psi:X\rightarrow X^{\prime}$ and a homeomorphism $\phi:B\rightarrow B^{\prime}$ sending $\Delta$ to $\Delta^{\prime}$ homeomorphically, such that $f^{\prime}\circ\psi=\phi\circ f$. We shall say that $\mathcal{F}$ is $(\psi,\phi)$-conjugate to $\mathcal{F}^{\prime}$ whenever the specification is required.

• (ii)

  If in addition $X$ and $X^{\prime}$ are symplectic manifolds and the fibrations are Lagrangian, we will say that $\mathcal{F}$ is symplectically conjugate to $\mathcal{F}^{\prime}$ if $\psi$ is a $C^{\infty}$ symplectomorphism and $\phi$ is a $C^{\infty}$ diffeomorphism.

• (iii)

  Given points $b\in\Delta$ and $b^{\prime}\in\Delta^{\prime}$, we shall say that $\mathcal{F}$ is (symplectically) conjugate to $\mathcal{F}^{\prime}$ over $\Delta$ (or over $b$ and $b^{\prime}$) if there are neighborhoods $U$ and $U^{\prime}$ of $\Delta$ and $\Delta^{\prime}$ (or of $b$ and $b^{\prime}$) respectively, such that $(f^{-1}(U),f,U)$ is (symplectically) conjugate to $((f^{\prime})^{-1}(U^{\prime}),f^{\prime},U^{\prime})$.