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You could also do the obvious generalization of quasimorphisms:

In the category of

Consider groups with a metric consider and look at maps maps $\phi: G \to H$ such that $d_H(\phi(g_1)\phi(g_2),\phi(g_1g_2))$ is bounded in some sense.

For instance , Lipschitz maps $\phi: (G,d_G) \to (H,d_H)$ such that $d_H(\phi(g_1)\phi(g_2),\phi(g_1g_2))$ is uniformly boundedforms , form a category.

Homogeneous quasimorphism quasimorphisms (maps $G \to \mathbb{R}$) are actual homomorphisms on abelian subgroups and are invariant under conjugation. By adjusting the above setup, perhaps one could get such features as well.

In the category of groups with a metric consider maps $\phi: G \to H$ such that $d_H(\phi(g_1)\phi(g_2),\phi(g_1g_2))$ is bounded in some sense.
For instance, Lipschitz maps $\phi: (G,d_G) \to (H,d_H)$ such that $d_H(\phi(g_1)\phi(g_2),\phi(g_1g_2))$ is uniformly bounded forms a category.