Isomorphism invariant metrics

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 $d:{\mathtt{C}}_0\times {\mathtt{C}}_0\to [0,\mathrm{\infty }]$ 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.