I feel, pacé Grothendieck, that the use of fundamental groupoid or $n$-groupoid for a strict structure in dimension $1$ and a weak structure in higher dimensions is confusing! But I seem to be out on a limb on this.
In any case, the higher dimensional strict structures have advantages for calculating homotopical invariants, as shown in my papers with Higgins and with Loday. These higher homotopy groupoid structures are known to be well defined for certain structured spaces, in fact for filtered spaces, and for $n$-cubes of spaces. This relates them well to classical homotopical structures, i.e. to relative homotopy groups, and to $n$-ad homotopy groups, and yield new calculations and understanding of of these.
I hope it will be useful to explain the route to this conclusion.
Philip and I defined a strict fundamental double groupoid $\rho(Y,X,x)$ of a pair of pointed spaces in 1974, but Frank Adams' opposition delayed the publication till 1978. I had been looking for a homotopy double groupoid of a space since 1965, but could not do it. In 1974 Philip and I agreed:
Whitehead's subtle theorem (1941-1949), proved using transversality and knot theory, that the crossed module $ \delta: \pi_2(X \cup _\lambda e^2 _\lambda,X,x) \to \pi_1(X,x) $ is free on the $2$-cells was an example of a universal property in $2$-dimensional homotopy theory;
If our conjectured $2$-dimensional van Kampen was true in some form, then it should imply Whitehead's theorem.
Whitehead's theorem involves relative homotopy groups.
We should therefore try to define a homotopy double groupoid in a relative situation.
Given we had little time remaining in Philip's stay at Bangor, we should try the simplest idea, and that seemed to be to consider homotopy classes of maps of a square $I^2$ into $Y$ which mapped the edges into the subspace $X$ and the vertices to the base point. Note that this a symmetric definition, and requires no choice of which edges to be mapped to the base point, as in the usual (and so unaesthetic?) definition of relative homotopy groups.
This was a sensible definition; the work that had already been done by then with Chris Spencer on double groupoids with connections, commutative cubes, and the precise relation of these double groupoids with crossed modules, allowed a proof of the van Kampen theorem and, and given work by Philip on induced crossed modules, a deduction of Whitehead's theorem, and indeed a generalisation to a theorem on $\delta:\pi_2(X\cup CA,X,x)\to \pi_1(X,x)$, with Whitehead's theorem being the case $A$ is a wedge of circles.
This suggests the advantage of a proper strategic analysis, even if delayed!
Note that one of the advantages of strict structures is that the calculation of colimits of them is reasonably clear; in particular, the calculation of colimits of crossed modules is an interesting extension of the calculation of colimits of groups. In algebraic topology, one use of homotopical invariants is to show that objects are not equivalent, and this needs precise answers.
It was then not hard to see the putative extension to filtered spaces; the work was in fact quite hard technically and conceptually, but was completed in 1977, with two CRAS notes published in that year, and the full papers in 1981.
Actually the interest of Whitehead's theorem seems not so much appreciated. It allows one, for the usual representation of the Klein bottle as an identification of a square $\sigma$, to write the more precise
$$\delta \sigma = a+b-a +b $$
instead of the usual $\partial \sigma = 2b$. See this presentation.
In 1982 Loday published a paper giving a definition of a fundamental cat$^n$-group of a pointed $n$-cube of spaces, and showed that such structures modelled all weak, pointed homotopy $n$-types. In a visit of mine to Strasbourg towards the end of 1981, I talked about my work with Philip Higgins, and together Loday and I conjectured a van Kampen type theorem for his cat$^n$-groups. This was proved by 1984 and published in 1987. It allows some quite new calculations of homotopical invariants, e.g. the homotopy $3$-type of $SK(G,1)$, the suspension of an Eileneberg-Mac Lane space.
These theorems have strong limitations, as is only to be expected, but they do allow new determinations and open out new prospects.
So one possible moral is that the concentration on bare topological spaces, without any further structure, is a more restricted endeavour. This fits with Section 5 of Grothendieck's "Esquisse d'un programme", where he argues that the needs of geometry require more than just a topology, and he argues for kinds of stratifications. One way of looking at this is to say that the specification of a space requires some kind of data, and that data will have some kind of structure; so, conjecturally, one should look for invariants of spaces with that kind of structure.
It would be interesting to compare these higher homotopy groupoid structures, and the weak structures in common current parlance, with the vision of the algebraic topologists of the early 20th century for higher dimensional versions of the nonabelian fundamental group. Since the finding in 1932 that Cech's definition of higher homotopy groups, which he submitted to the ICM at Zurich, led to abelian groups, this has seemed to be a mirage.
