Consider the following two structure-adding refinements of the fundamental group of a topological space:
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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.
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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!
Best Answer
I figured my comments are getting too long so I should put some of them here, even though I'm using a different topology on $\pi_1$ (often it is the same, but certainly not always) and a possibly different pro-group (which I can't tell as I'm not familiar with toposes). EDIT: Unless otherwise noted, I'm assuming that $X$ is compact metrizable to be on the safe side.
The relationship goes in 2 steps. First the fundamental pro-group can be compared with the Cech fundamental group endowed with the inverse limit topology. In particular, as noticed by Atiyah and G. Segal, they contain precisely the same information as long as the fundamental pro-group is Mittag-Leffler. For other results in this direction see lemma 3.4 in "Steenrod homotopy". The topological homomorphism between $\pi_1(X)$ (with topology as in my comment) and the topological Cech fundamental pro-group is discussed in theorem 6.1, section 5 and elsewhere in "Steenrod homotopy".
To summarize the relationship very roughly, $\pi_1(X)$ topologized as in my comment retains much of the inverse limit of the fundamental pro-group, discards all of its derived limit, but instead gets something of the derived limit of the second homotopy pro-group (exactly how much is still a subject of ongoing research, see Theorem 6.5 and remark to corollary 8.8 in "Steenrod homotopy"). Still this is not all that it contains (cf. example 5.7 in "Steenrod homotopy").
Added later: Addressing Mike's comment, let me elaborate on the choice of topology on $\pi_1$.
The Mathoverflow thread on the quotient topology on $\pi_1$, and particularly Andrew Stacey's answer there, make it clear that the only reason that this topology is not compatible with multiplication is that the product of two quotient maps need not be a quotient map. But wait, the product of two quotient maps is a quotient map in the category of uniform spaces and uniformly continuous maps! See Isbell's "Uniform spaces" (1964), Exercise III.8(c). In fact, the product of any (possibly infinite) collection of quotient maps is a quotient map in the uniform category. (Quotient maps are defined in any concrete category over the category of sets, as explained e.g. in The Joy of Cats.)
This means that the topology of the quotient uniformity on $\pi_1(X)$ makes it into a topological group. By this topology I mean the following. If $X$ is compact, it carries a unique uniformity, and then the space of continuous (=uniformly continuous) maps $(S^1,pt)\to (X,pt)$ is a uniform space (endowed with the uniformity of uniform conergence; if $X$ is metrizable by a metric $d$, this uniformity is metrizable by the metric $D(f,g)=\sup\limits_{x\in X}\ d(f(x),g(x))$.) If $X$ is not compact, then we need to fix some uniformity on it and the above works. To what extent the resulting topology on $\pi_1(X)$ depends on the choice of uniformity is a good question.
Added still later: With this in mind, I no longer see a good reason for myself to think at all about the quotient topology on $\pi_1(X)$. As you can see for instance from my comment here, I have good reasons to believe that quotient topology is a sensible notion only when it coincides with the topology of quotient uniformity (i.e. for quotients of compact spaces), and otherwise the topology of quotient uniformity is the way to go. The present situation does not seem to be an exception.
Now does the topology of the quotient uniformity coincide with the "inverse limit" topology, at least when $X$ is compact? I find it easier to go down one dimension, and compare the two similar topologies on $\pi_0(X)$, the set of path-components. The "inverse limit" topology is again induced from the inverse limit topology on the Cech $\pi_0$ (which is the inverse limit of $\pi_0(P_i)$, where $X$ is the inverse limit of the compact polyhedra $P_i$), and alternatively its basic open sets are those collections of path components whose unions are the point-inverses of maps of $X$ to finite sets. And as long as $X$ is compact (or is endowed with a uniformity), we have the topology of the quotient uniformity, viewing $\pi_0(X)$ as the quotient of $X$ by the equivalence relation of being in the same path-component.
Firstly let me say that (often or not) the two topologies do share something. For instance if $X$ is the $p$-adic solenoid then both are anti-discrete, for apparently very different reasons: the inverse limit topology simply because $X$ is connected; and the topology of the quotient uniformity is antidiscrete because every point lies in the closure of the path-component of every other point. If you take a wedge of two solenoids or even a "garland" of solenoids obtained from their disjoint union by identifying pairs of points taken from distincts solenoids, both topologies are still antidiscrete. In the case of the topology of the quotient uniformity, this is so because every two points $x,y$ are the endpoints of a chain $x=p_0,\dots,p_n=y$ such that each $p_{i+1}$ lies in the closure of the path-component of $p_i$.
On the other hand, let us do the pseudo-arc. Its path-components are single points, so $\pi_0(X)=X$. The topology of the quotient uniformity is the original topology of $X$; and the "inverse limit" topology is anti-discrete because $X$ is connected.
I haven't really thought about the two topologies on $\pi_1$ but I see no reason why this should be terribly different from the case of $\pi_0$.
Still more: While the relation of the two topologies on $\pi_1$ might be not so easy to comprehend, the topology of the quotient uniformity on $\pi_1$ relates in a seemingly more comprehensible way directly to the topologized Steenrod $\pi_1$ (whose relation to the fundamental pro-group is clear). The Steenrod $\pi_1(X)$ can be defined as $\pi_1(holim P_i)$, where the compact metrizable space $X$ is the inverse limit of the polyhedra $P_i$ and $P_0=pt$. (If $P_0\ne pt$ we can shift the indices by one and add a new $P_0$, namely $pt$.) If you think about the construction of $holim$, you will see that $\pi_1(holim P_i)$ is the set of equivalence classes of level-preserving base-ray-preserving maps $S^1\times [0,\infty)\to P_{[0,\infty)}$ under the relation of level-preserving base-ray-preserving homotopy, where $P_{[0,\infty)}$ denotes the mapping telescope of the inverse sequence $\dots\to P_1\to P_0$ (glued out of the mapping cylinders $P_{[i,i+1]}$ of the individual bonding maps). Since $P_0=pt$, "level-preserving" can in fact be replaced by "proper" (see Lemma 2.5 in "Steenrod homotopy"). In other words, the Steenrod $\pi_1(X)$ is the set of equivalence classes of base-ray preserving maps from $D^2$ to the one-point compactification $P_{[0,\infty)}^+$ of the mapping telescope such that the preimage of the point at infinity is precisely the center of the disk (and nothing else).
Now it is not hard to see that the topology on the Steenrod $\pi_1(X)$ (induced from the inverse limit topology on the Cech $\pi_1(X)$ ) is precisely the topology of the quotient uniformity. The uniformity is on a subspace of the space of all continuous (=uniformly continuous) maps from $D^2$ to $P_{[0,\infty)}^+$, and so it is the subspace uniformity (of the uniformity of uniform convergence).
The continuous map from $\pi_1(X)$ with the topology of the quotient uniformity to the Steenrod $\pi_1(X)$ is given by Milnor's lemma (see Lemma 2.1 in Steenrod homotopy): every map $S^1\to X$ extends to a level-preserving map $S^1\times [0,\infty]\to P_{[0,\infty]}$, where $P_{[0,\infty]}$ is Milnor's compactification of the mapping telescope by a copy of $X$, and this extension is well-defined up to homotopy through such extensions. Also close maps have close extensions (from the proof of Milnor's lemma). The quotient $P_{[0,\infty]}/X$ is homeomorphic to $P_{[0,\infty]}^+$, so the close extensions descend to close representatives of elements of the Steenrod $\pi_1$.