The fundamental groupoid \(\Pi_1(X)\) of a space \(X\) is a groupoid whose objects are the points of \(X\) and whose morphisms are endpoint homotopy equivalence classes of paths \(x\to y\).
\(\mathrm{Ob}(\Pi_1(X)) = X\)
\(\mathrm{Hom}_{\Pi_1(X)}(x,y) = \{ x \to y \}/\sim\)
There are some obvious properties:
\(\mathrm{End}_{\Pi_1(X)}(x) = \pi_1(X,x)\), the fundamental group of \(X\) at \(x\).
\(\pi_1(X,x) \subset \Pi_1(X)\) is a categorical equivalence if \(X\) is path connected.
For a small category \(\mathcal{D}\), a functor \(F:\mathcal{D}\to \mathcal{C}\) is also called a \(\mathcal{D}\)-shaped diagram in \(\mathcal{C}\).
Colimit of a \(\mathcal{D}\)-shaped diagram \(F:\mathcal{D}\to \mathcal{C}\) is an object \(\mathrm{colim}F\) with a natural transform \(\iota: F \Rightarrow \underline{\mathrm{colim}F}\) (here the underline denotes a constant functor) that factor through uniquely any other natural transform into a constant functor \(F\Rightarrow \underline{A}\).
Limit of a \(\mathcal{D}\)-shaped diagram \(F:\mathcal{D}\to \mathcal{C}\) is an object \(\lim F\) with a natural transform \(\pi: \underline{\lim F} \Rightarrow F\) that factor through uniquely any other natural transform from a constant functor \(\underline{A} \Rightarrow F\).
When \(\mathcal{D}=\{a \leftarrow b \rightarrow c\}\), the colimit is a pushout.
When \(\mathcal{D}=\{a \rightarrow b \leftarrow c\}\), the limit is a pullback.
When \(\mathcal{D}=\{a \to\to b\}\), the limit is an equalizer and the colimit is a coequalizer.
A category is complete if it has all small limits, and cocomplete if it has all small colimits.
\(\mathcal{C}\) is complete if and only if \(\mathcal{C}\) has all equalizers and products.
\(\mathcal{C}\) is cocomplete if and only if \(\mathcal{C}\) has all coequalizers and coproducts.
Let \(\mathcal{O}\) be a category of a open covering of \(X\) (has finite intersections) whose morphisms are inclusions. Let
\[\Pi_1|\mathcal{O}: \mathcal{O}\to \mathcal{G}\mathcal{P}\]
be the \(\mathcal{O}\)-shaped diagram in \(\mathcal{G}\mathcal{P}\) (the category of groupoids), mapping \(U\mapsto \Pi_1(U)\) and inclusions \(V\subset U\) to the obvious maps \(\Pi_1(V)\to \Pi_1(U)\). Then we have a canonical isomorphism of groupoids:
\[\Pi_1(X) \cong \mathrm{colim}_{\mathcal{O}}\, \Pi_1|\mathcal{O}: \mathcal{G}\mathcal{P}.\]
To show this, using the universal property of colimits, for any natural transform \(\Pi_1|\mathcal{O}\Rightarrow \underline{\mathcal{G}}\) to a constant groupoid functor \(\underline{\mathcal{G}}: \mathcal{O}\Rightarrow \mathcal{G}\mathcal{P}\), we need to show that it factor uniquely through the constant groupoid functor \(\underline{\Pi_1(X)}\).
\[\eta:\Pi_1|\mathcal{O}\Rightarrow \underline{G} \text{ factor through }\underline{\Pi_1(X)}.\]
It is obvious that \(\Pi_1|\mathcal{O}\) maps into \(\underline{\Pi_1(X)}\) in an canonical way, so we need to specify how to define \(\underline{\Pi_1(X)}\Rightarrow \underline{\mathcal{G}}\) or equivalently, a functor \(\Pi_1(X)\to \mathcal{G}\), using the information in \(\eta\).
The data of this natural transform consists of morphisms for each open set \(U\in \mathcal{O}\), a functor \(\eta_U:\Pi_1(U)\to \mathcal{G}\). The compatible family of functors \(\eta_U\) already determined the image of each point \(x\in \Pi_1(X)\) in \(\mathcal{G}\).
For the morphisms in \(\Pi_1(X)\),
If the morphism \(x\to y\) is represented by a path that lies completely in one of the \(U\in \mathcal{O}\), then the path lies in \(\Pi_1(U)\) and \(\eta_U\) gives the image of this morphism in \(\mathcal{G}\).
If the morphism lies also in other \(U'\), then they lie in a finite intersection, for which compatibility guarantees that the image in \(\mathcal{G}\) is the same.
Otherwise, break the path into segments \(x_0\to x_1\to \dots\to x_n\), each segments lies in some \(U_i\in\mathcal{O}\). The image of each segment \(x_i\to x_{i+1}\) is given by \(\eta_{U_i}:\Pi_1(U_i)\to \mathcal{G}\), they have a composition in \(\mathcal{G}\). We define the image of the path \(x_0\to x_n\) to be this composed element.
To show that when we have two homotopic paths \(f_0, f_1: x\to y\) with their own choice of segments, they define the same image in \(\mathcal{G}\).
Consider dividing the square \(I\times I\). Let the upper side \(I\times 0\) contain the first division into segments and the lower side \(I\times 1\) contain the second division. Since \(I\times I\) is compact, it is possible to refine each divisions so that we can break \(I\times I\) into rectangles, each rectangle lies completely in at least one \(U\in\mathcal{O}\).
Each rectangle now gives a local commutative diagram of two pairs of paths in \(\Pi_1(U_\bullet)\), whose two images in \(\mathcal{G}\) are the same. Putting all of the commutativity together gives us \(f_0\) and \(f_1\) have the same image in \(\mathcal{G}\).
Theorem (Classical Van Kampen Theorem)
Let \(X\) be a path connected and choose a basepoint \(x\in X\). Let \(\mathcal{O}\) be a category of open cover of \(X\) by path connected open subsets \(U\ni x\) closed under finite intersections. The functor \(\pi_1(,x)\) restricted on \(\mathcal{O}\) gives an \(\mathcal{O}\)-shaped diagram \(\pi_1|\mathcal{O}:\mathcal{O}\to \mathcal{G}\) in the category of groups. Then the fundamental group \(\pi_1(X,x)\) is isomorphic to the colimit of this diagram:
\[\pi_1(X,x) \cong \mathrm{colim}_{\mathcal{O}}\, \pi_1|\mathcal{O}: \mathcal{G}.\]