Consider the following two structure-adding refinements of the fundamental group of a topological space:

• the set $\pi_1(X)$ inherits a quotient topology from the compact-open topology of $X^{S^1}$, under which it is sometimes a topological group. This was discussed here.

• the topos of sheaves on X has a fundamental group, which is in general a pro-group, reducing to an ordinary group if X is locally simply connected.

Pro-groups and topological groups are not unrelated concepts; in particular, both have a common "generalization" to localic groups. Are there any known relationships between the "topological" and "toposophic" fundamental groups of a space? Do they capture similar or different information?