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added to van Kampen theorem a clean statement for the group-version
In light of the discussion around higher homotopy van Kampen theorems, I have expanded van Kampen theorem to distinguish more clearly the homotopy version from the strict version, and included a proof of how to get from the homotopy version to the strict one. This is kind of like the proof I gave on MO for how to get the classical strict version out of something like Lurie’s version, but simpler (one doesn’t have to introduce higher toposes if all one wants is a statement about 1-groupoids!); it’s basically what is done in Dror Farjoun’s paper, cleaned up with some model-category language. I think Dror isn’t fully explicit about how the set of basepoints is dealt with, although he does discuss them at least in the context of the 0-skeleton of a complex in the published version (which is different from the draft version which used to be the one linked on the page).
Pasted here an old discussion from the page:
Ronnie This paper does not seem to mention the fundamental groupoid on a set of base points. Is there a version of the result on homotopy colimits for many base points?
The original idea for many base points was to calculate the fundamental group of the circle via a van Kampen type theorem for non connected spaces: this required the many base point version.
It seems useful to use $\pi_1(X,X_0)$ where $X_0$ is chosen according to the geometry at hand, usually somewhere between a single point and the whole space. Grothendieck agreed!
Arguments for (and against!) groupoids are more fully set out on http://www.bangor.ac.uk/r.brown/gpdsweb.html.
Mathieu I don’t think one need to use a set of base points in the case of homotopy colimits, since in this case we work up to equivalences of groupoids. If you apply $\pi_1$ to the circle presented as the homotopy pushout of the map $2\to 1$ along itself (where $2$ is the discrete space on two elements), you get a groupoid equivalent to the (bicategorical) pushout in the 2-category of groupoids of $2\to 1$ along itself (this time, seen as discrete groupoids), which is (up to equivalence) the group $\Z$ seen as a one-object groupoid.
Ronnie This all seems more complicated than the statement: the group $\Z$ is up to isomorphism obtained from the unit interval groupoid $I$ by identifying 0 and 1, in the category of groupoids. Analogous lower dimensional identifications become more significant in the applications of higher homotopy van Kampen theorems, which allow for some computations for example of homotopy 2-types, not so far obtained by homotopy colimit methods. These require higher homotopy groupoids for their proof.
The many base point case is used in proving subgroup theorems in group theory. Higgins also gave in 1976 a very nice normal form for the fundamental groupoid of a graph of groups; since a graph has vertices, is is not surprising that this groupoid has the same vertices as the graph. This elegant idea has been ignored by the experts in that area.
I agree that homotopy colimits are interesting. For example, I like to consider the trefoil groupoid $T$ which is the homotopy pushout in the category of groupoids of the two maps $\Z \to \Z$ given by multiplication by 2 and by 3. There are advantages in keeping this with two objects, as reducing to the trefoil group loses some structure, such as the distinction between two generators.
Similarly, it is convenient to consider $\pi_1(\Delta^n, \Delta^n_0)$, the fundamental groupoid of the $n$-simplex on its set of vertices. This keeps the geometry of the simplex. So the nerve of a groupoid $G$ is the simplicial set which in dimension $n$ is $Gpd(\pi_1(\Delta^n, \Delta^n_0),G)$.
For all I know, there may be advantages in replacing loop space theory by a many-pointed theory, involving the structures which arise from considering all paths, and even all cubes, between the base points!
Quickly reducing a groupoid to one object is to me a bit like always choosing a basis for a vector space.
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