Fibrewise Orthogonal Morphisms

A pair of morphisms $e:{{A}\to{B}},m:{{X}\to{Y}}$ are said to be fibrewise orthogonal, denoted $e\fperp m$ if every Base Change $e^{*}$ of $e$ is Orthogonal to $m$.

Explicitly, this means that for every $u$ fitting into a diagram

the resulting $u^{*}(e)$ is Orthogonal to $m$.

Following the terminology for Orthogonal Morphisms, we shall say that $e$ is fibrewise left orthogonal to $m$, and that $m$ is fibrewise right orthogonal to $e$. We can further generalize this terminology to a class of morphisms $\mathcal{A}$, and use $\mathcal{A}^{\fperp}$ to denote the class of morphisms that are fibrewise right orthogonal to every map in $\mathcal{A}$, and ${}^{\fperp}\mathcal{A}$ to denote the class of morphisms that are fibrewise left orthogonal to every map in $\mathcal{A}$.