As far as I understand, there are several ways of defining $\infty$-categories. One of them is to think of $\infty$-cateogries as $top$-enriched categories. Hence we can think of $\infty$-groupoids as generalizing topological groups. Functors between groupoids are the generalization of group homomorphisms. Hence my question is if $\infty$-functors of $\infty$-groupoids generalize continous group-homomorphisms? For instance, if $G,H$ are topological groups, and $BG,BH$ denote the associated topological groupoid/$\infty$-groupoids, do we have
$$\text{Hom}\_{\infty\text{-gpd}}(BG,BH)\cong \text{Hom}\_{cont}(G,H)?$$



Morphisms of $\infty$-groupoids
Only for special kinds of topological groups (like Lie groups or groups with an underlying topological manifold). This fails horribly for e.g. profinite groups, which are detected as discrete groups by the topological nerve.


