  Within a category $\mathtt{C}$, having objects $\mathtt{C}_0$, it may be
instructive to know not only that two objects are non-isomorphic, but also how
far from being isomorphic they are. We introduce pseudo-metrics $d:\mathtt{C}_0
\times \mathtt{C}_0 \to [0,\infty]$ with the property that $x\cong y$ implies
$d(x,y)=0$. We also give a canonical construction that associates to each
isomorphism invariant a pseudo-metric satisfying that condition. This
guarantees a large source of isomorphism invariant pseudo-metrics. We examine
such pseudo-metrics for invariants in various categories.
