The Seifert-van Kampen theorem is a classical theorem in algebraic topology which computes the fundamental group of a pointed topological space in terms of a decomposition into open subsets.
It is most naturally expressed by saying that the fundamental groupoid functor preserves certain colimits. Here there is a bifurcation in possible generalizations, however. The colimits of spaces we consider (covers by open subsets) are both strict colimits and homotopy colimits. But we can ask the resulting colimit of groupoids to be either a homotopy colimit (i.e. a 2-colimit) or a strict colimit. The first is more natural for formal work; the second is more useful for computation. Accordingly, we will speak of homotopy van Kampen theorems and strict van Kampen theorems.
The most classical version of the Seifert-van Kampen theorem for the fundamental group is the following.
Let $X$ be a topological space covered by open subsets $U,V \subset X$ such that $U \cap V$ is path connected. Then for every choice of basepoint $x \in U \cap V$, the diagram of homotopy groups
We can regard the classical theorem as being a statement about fundamental groupoids as follows. Recall that any group $G$ can be considered as a 1-object groupoid $\mathbf{B}G$ by delooping. It is easy to see that the functor $\mathbf{B}$ takes colimits of groups to strict colimits of groupoids. Moreover, a strict pushout of one-object groupoids is also a 2-pushout, because the maps we are pushing out are bijective on objects, hence cofibrations in the canonical model structure on $Gpd$. Thus, under the hypotheses of the classical van Kampen theorem, we have a square
which is both a strict pushout and a 2-pushout in $Gpd$.
Now let $X'$ be the path component of $X$ which contains $U\cap V$ (there is exactly one such, because $U\cap V$ is path-connected), and similarly $U'$ and $V'$. Then the fundamental groupoid $\Pi_1(X')$ is equivalent to $\mathbf{B}\pi_1(X,x)$, and similarly in the other cases. Since 2-pushouts are invariant under equivalence of groupoids, we also have a 2-pushout square
This is an instance of what we referred to above as a homotopy van Kampen theorem. But we can easily make it a strict van Kampen theorem as well.
Note that the functors $\Pi_1(U\cap V) \to \Pi_1(U')$ and $\Pi_1(U\cap V) \to \Pi_1(U')$ are injective on objects, hence cofibrations in the canonical model structure, and so their strict pushout is also a 2-pushout. This strict pushout is therefore equivalent to $\Pi_1(X')$. (It need not be isomorphic to it, but this can be remedied by defining the fundamental groupoid on a set of specified basepoints as below.)
We now consider more general versions of the van Kampen theorem which do not require $U\cap V$ to be path-connected, and which compute all of $\Pi_1(X)$ rather than just $\Pi_1(X')$.
Let us define $\Pi_1(X,X_0)$ of a space $X$ and a subset $X_0 \subseteq X$ to be the full subgroupoid of $\Pi_1(X)$ on the set $X_0$. More generally, if $U$ is given as a subspace of $X$ and $X_0 \subseteq X$, we write $\Pi_1(U,X_0)$ for $\Pi_1(U,X_0\cap U)$.
If $X_* =(X,X_0)$ is a pair consisting of a space $X$ and a set $X_0$ of base points, we say $X_*$ is connected if $X_0$ meets each path component of $X$. This is clearly a necessary and sufficient condition for $\Pi_1(X,X_0)$ to be equivalent to $\Pi_1(X)$.
Now let $\{U^i \to X\}$ for $i \in I$ be an open cover of $X$. If $d=(i,j) \in I^2$ we write $U^d$ for $U^i \cap U^j$, and similarly for $n$-fold intersections. For any $n$-tuple $e\in I^n$, let $U^e_*$ be the pair $(U^e,X_0 \cap U^e)$. We then have a coequaliser diagram of pairs of spaces, where $a,b,c$ are determined by inclusions:
If the pairs of spaces $U^f_*$ are connected for all 1-, 2-, or 3-fold intersections $U^f_*$ of the pairs $U^i_*$, then
(Conn) The pair $X_*$ is connected; and
(Iso) The fundamental groupoid functor $\Pi_1$ takes the above coequaliser diagram of pairs of spaces to a (strict) coequaliser diagram of groupoids.
The limit “3” on the connectivity hypothesis may be analyzed as “$1+2$”, where $1$ is the categorical dimension of the fundamental groupoids we are considering and $2$ is a universal constant. This becomes evident when generalizing to higher homotopy van Kampen theorems.
In one proof of the theorem (due to Brown and Razak, see below), the number 3 arises from the Lebesgue covering dimension of $\mathbb{R}^2$. This proof verifies the universal property for a coequaliser, using basic techniques: subdivide a path; deform a subdivision so that it is product of paths joining points of $X_0$; subdivide a homotopy rel end points, and deform this subdivision so that all subpaths join points of $X_0$; any composition of commutative squares in a groupoid is commutative.
In another, more categorical proof explained in the next section, the number $3$ arises from the fact that 2-colimits can be computed using coproducts and descent objects of 2-truncated simplicial objects.
Note that the theorem about groupoids does not reduce to a theorem about groups. For example, $X$ may be a connected space which is the union of two open sets each with many components and whose intersection has many path components. Such examples arise commonly from applications to combinatorial group theory; consider for example a covering space of a wedge of spaces. Or the connected $X$ may be the union of 23 open sets whose three fold intersections have 123 path components. In each case the fundamental group one might want to calculate is in the middle of this complicated combinatorial situation, but at least the theorem has turned a topological problem into a group theory and combinatorial problem, and the remarkable fact is that the fundamental groups are completely determined by the theorem. This is an “anomaly” for traditional algebraic topology, where invariants relating adjacent dimensions may be determined by exact sequences which do not give complete information. The reason for the success seems to be that groupoids have structure in dimensions 0 and 1, and so can model the gluing of spaces.
This suggests that higher homotopy van Kampen theorems could give new kinds of homotopical information, i.e. colimit theorems for higher homotopy invariants, which has proved to be so.
From a 2-categorical point of view, the strict van Kampen theorem above can be decomposed into the following parts:
When $X$ is the union of a family of open sets, then it is the homotopy colimit of a diagram involving these open sets.
The fundamental groupoid functor preserves homotopy colimits (i.e. takes them to homotopy colimits, or equivalently 2-colimits, of groupoids).
Certain 2-colimits of groupoids can be computed over “2-truncated” subdiagrams.
In certain situations, strict colimits of groupoids are also 2-colimits.
We can then prove the theorem by proving each of these three parts separately. This is essentially accomplished in Farjoun.
If a space $X$ is the union of a family of open sets $(U^i)_{i\in I}$, then we have a weak equivalence
where $C_\bullet(U)$ is the Cech cover of $X$, a simplicial space defined by
See Dugger-Isaksen 01.
Suppose $I$ is a small category and $(X^i)_{i\in I}$ is a diagram of spaces. Then we have an equivalence of groupoids
The functor $\Pi_1$ is, up to equivalence, the composite of itself with the 1st Postnikov approximation functor $P_1 : Top \to Top$. We can regard $P_1$ as a Bousfield localization functor for $Top$ (for localizing at the map $S^2 \to \ast$), so it preserves homotopy colimits. And $\Pi_1$ is part of a Quillen equivalence between $Gpd$ and the 1-localized model structure on $Top$, so it also preserves homotopy colimits.
If $X$ is the union of $(U^i)_{i\in I}$, and $X_0\subseteq X$ is a set of basepoints which intersects each 1-, 2-, and 3-fold intersection of opens $U^i$, then $\Pi_1(X,X_0)$ is equivalent to the 2-colimit (i.e. codescent object) of the following diagram
If we neglect the $X_0$‘s, then what we have above is the 2-truncated part of the homotopy colimit of groupoids obtained by applying Theorem 2 to Theorem 1. Since homotopy colimits of simplicial objects in $Gpd$ are equivalent to codescent objects of their 2-truncations (essentially because groupoids are 1-groupoids), this has an equivalent homotopy colimit. Finally, the assumption on $X_0$ ensures that restricting to it as a set of basepoints does not change the equivalence type of any of the groupoids in the 2-truncated diagram.
Under the hypotheses of Theorem 3, we have that $\Pi_1(X,X_0)$ is isomorphic to the strict coequalizer of
Let $A = P(I)$ be the power set of the indexing set $I$, regarded as a poset, and consider the functor $F:A\to Gpd$ defined by $F(J) = \Pi_1(\bigcap_{i\in J} U^i, X_0)$. We can verify that the 2-colimit of $F$ is equivalent to the codescent object in Theorem 3, for essentially the same reasons cited there. (Formally, one can say that the functor $P_{\le 3}(I) \to A$ is “2-final”.)
However, $F$ is cofibrant in the projective model structure on $Gpd^A$, which is to say that the functor $A\xrightarrow{F} Gpd \xrightarrow{ob} Set$ is free (relative to the forgetful functor $Set^A \to Set^{ob(A)}$): its generators are the points $x\in F(J)$ such that $J = \{ i\in I \,|\, x\in U^i \}$. Thus, the 2-colimit of $F$ is equivalent to its strict colimit, which in turn is easily seen to be the coequalizer shown above.
This shows that $\Pi_1(X,X_0)$ is equivalent to the strict coequalizer shown above. But this equivalence is clearly also bijective on objects, so it is an isomorphism of groupoids.
Let $X=S^1$, $U= S^1 \setminus \{N\}$, and $U= S^1 \setminus \{S\}$, where $N$ and $S$ are antipodal points. Then $U\cong V\cong (0,1)$ are contractible, while $U\cap V \cong (0,1) \sqcup (0,1)$. Thus the homotopy van Kampen theorem gives a 2-pushout of groupoids that is equivalent to
It’s not hard to compute that this 2-pushout must be $\mathbf{B}\mathbb{Z}$. But we can also obtain this directly from the strict van Kampen theorem with a set $X_0 = \{u,v\}$ of two basepoints with $u\in U$ and $v\in V$, which yields a strict pushout of groupoids
where $\mathcal{I}$ is the invertible interval category, which is (literally) $\Pi_1(U,X_0)$ and $\Pi_1(V,X_0)$.
Note that the version of the theorem for fundamental groups does not apply, since $U\cap V$ is not path-connected.
See also this mathoverflow discussion
The following paper gives the general theorem for the fundamental groupoid on a set of base points:
The pushout theorem is among applications of groupoids to $1$-dimensional homotopy theory and to the Jordan Curve Theorem in:
Ronnie Brown Topology and Groupoids , Booksurge (2006). Available from amazon.
The proof for pushouts is discussed in some detail in
as a background to the following chapters on versions for crossed modules and crossed complexes, i.e. for higher dimensional versions.
The proof via homotopy colimits is roughly taken from
Other references cited above: