Let's Grothendieck Everything in Sight (Week 3)

If you're following along, last time I showed you how to construct a category of topological pairs using the Grothendieck construction. This week I'm going to Grothendieck another functor from $ \mathsf{Top}$ to $ \mathsf{Cat}$; the fundamental groupoid functor.

Let $ \pi_1 : \mathsf{Top} \to \mathsf{Cat}$ be the functor defined as follows.

• For a topological space $ X$, $ \pi_1(X)$ is a category where,
  • an object is an element $ x, y, \cdots $ of $ X$ and,
  • a morphism from $ x$ to $ y$ is a continuous path $ \gamma : [0,1] \to X$ with $ \gamma(0) =x$ and $ \gamma(1)=y$. I will denote this path by $ \gamma: x \to y$.
  • To compose a path $ \gamma: x \to y$ with $ \beta: y \to z$ you walk along the paths in sequence...but twice as fast so it's still a map from $ [0,1] \to X$. I will denote this composite by $ \beta \cdot \alpha$.
• For a continuous function $ f: X \to Y$, $ \pi_1(f): \pi_1(X) \to \pi_1(Y)$ is a functor which sends
  • objects $ x \in \pi_1(X)$ to $ f(x)$ and,
  • morphisms $ \gamma: [0,1] \to X$ to the composite $ f \circ \gamma$

Okay now let's integrate. Following the definition from before, $ \int \pi_1$ is a category where

• objects are pairs $ (X, p)$ where $ X$ is a topological space and $ p$ is a point in $ X$ and,
• a morphism $ (f, \gamma): (X,p) \to (Y,q)$ is a continuous function $ f: X \to Y$ and a path $ \gamma : f(p) \to q$.
• To compose a morphism $ (f, \gamma) : (X,p) \to (Y,q)$ with a morphism $ (g, \beta) :(Y, q) \to (Z,r)$ you first use $ g$ to map $ \gamma$ to a path $g \circ  \gamma$ in $ Z$ and then compose it with $ \beta$ to get a path $ \beta  \cdot (g \circ \gamma)$. The composite morphism will be the pair $ (g \circ f, \beta \cdot (g \circ \gamma) )$.

Ok so now let's take step back and think about the Eldritch monster we've created. The objects in $ \int \pi_1$ are pointed topological spaces...but the morphisms are different. These morphisms don't preserve basepoints so this is not the category of pointed topological spaces. However, they do preserve basepoints up to a path.

This is really nice because often in topology the choice of basepoint can seem rather arbitrary. For a space $ X$ you can compute the fundamental groups $ \pi_1(X,x)$ and $ \pi_1(Y,y)$ for two distinct points $ x$ and $ y$ in $ X$. These two groups are only different (not isomorphic) if $ x$ and $ y$ live in different path components of $ X$. Our Grothendieck construction takes this into account by allowing change of basepoint in our morphisms.

In my opinion the most opaque part of the Grothendieck construction is the definition of composition. So another reason to care about this category is that it gives a nice visual intuition for this composition.

Puzzle: The Grothendieck construction of $ \pi_1$ is closely related to the Grothendieck construction of $ \mathsf{id} : \mathsf{Cat} \to \mathsf{Cat}$. Describe this Grothendieck construction.