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?

Edit: As Theo pointed out in a comment, one difference is that the toposophic fundamental group is defined in terms of coverings rather than paths. This makes it better-behaved for non-locally-path-connected spaces, where there may not be very many maps out of $S^1$. But if the space is "nice" enough so that there are "enough paths to retain all the information about coverings," we can still ask about comparing the quotient topology on the group of paths with the pro-structure on the group of coverings. It might be that imposing that "niceness" condition already trivializes the extra information in one or the other or both, but if so then that would be a good answer to the question!