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The fundamental groupoid of a topological space $X$, $\pi_{\le 1}(X)$ is a category where the objects are the points of $X$ and the morphisms homotopy classes of paths from any two points of $X$. Now, supposedly, $\pi_0(X)$ is the set of isomorphism classes of objects in the groupoid. Can someone help me unpack that last statement? Does it simply mean that two points are isomorphic if they're path connected?

Edit: Based on the comments and answer I seem to have the general idea; however something seems odd to me, we don't seem to gain any information from by considering $\pi_{\le 1}(X)$ as opposed to $\pi_{0}(X)$ (i.e. either consideration will tell you about the connectedness of $X$). Am I wrong?

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Yes, that's precisely what it means. In particular, if $\gamma$ is a path from $x_0$ to $x_1$ then the isomorphism is $\gamma$ itself. If $X$ is path connected, then there's one isomorphism class and $\pi_0(X)$ has it's usual definition, since a group is just a groupoid with a single object.

Edit: So I noticed that I actually made a bit of a mistake above; I mixed up notation. Here is what $\pi_0$ and $\pi_1$ are typically defined as:

  • $\pi_0(X)$ is the set of path components of $X$

  • If $x_0\in X$, then $\pi_1(X,x_0)$ is the group of homotopy classes of loops based at $x_0$, and in particular if $X$ is path-connected then we call this $\pi_1(X)$, because the base point $x_0$ does not matter.

I never took a course that specifically used the "groupoid" approach , but it seems that the benefit in that it gives you sufficient generality in that you can recover both $\pi_0$ and $\pi_1$ from the groupoid (hence the notation $\pi_{\le1}(X)$).

That is, if we define $\pi_0(X)$ to be the isomorphism classes of $\pi_{\le1}(X)$, then this is just the set of path components, because any path is an isomorphism as we mentioned above. So this agrees with the first definition. And if $x_0$ is a point of $X$, then considered as an object of $\pi_{\le1}(X)$, we can define its group of automorphisms to be $\pi_1(X,x_0)$ (which matches our previous definition again. If this is not immediately clear, think about why this is!).

And in particular, if the space is path connected, then all points are isomorphic, and $\pi_1(X,x_0)\cong\pi_1(X,x_1)$ (where this isomorphism is conjugation by $\gamma$ for any $\gamma$ connecting the points) for all $x_0,x_1\in X$, and we just call this group $\pi_1(X)$.

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    The isomorphic is actually $\gamma$ itself (not conjugation by $\gamma$). In $\pi(X)$ the morphisms are homotopy classes of paths, and all morphisms are isomorphisms. – 2017-01-26
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    @NajibIdrissi good catch, my mistake. – 2017-01-26
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    @AlexMathers I hope you don't mind but I expanded my question a little. Details are contained in the "Edit:" – 2017-01-26
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    @Bob see my edit. – 2017-01-26
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Regarding your edit: the isomorphism classes of objects are indeed given by $\pi_0(X)$. But given a class $[x] \in \pi_0(X)$, you can also consider the automorphisms of $x$ in $\pi_{\le 1}(X)$. This is a group, and it is isomorphic to $\pi_1(X, x)$. So there's more information than just $\pi_0(X)$.

Basically $\pi_{\le 1}(X)$ is a way of packing all the information about path-connected components and fundamental groups by taking care of base points in a coherent way. You can't talk about "the fundamental group of $X$" because you need a base point, but you can talk about the fundamental groupoid of $X$.