Fundamental group as functor I have read  little bit category theory on my own. These days I am reading Algebraic Topology from Hatcher's.
I guess fundamental group can be thought of as functor between category of topological spaces and category of groups.
What are other nice properties of this functor?
We know fundamental group of $X\times  Y$ is the product of fundamental group of $X$ and $Y$. How can we describe this property in terms of category theory.
Also, It is possible for two spaces to be non homeomorphic, but still having same fundamental group. Can we express this property in terms of category theory language. 
 A: I would say, that the fundamental group is a functor between pointed topological spaces of the from $(X,x_0)$ and groups. The morphisms of the first category are pointed maps $f\colon (X,x_0)\to (Y,y_0)$. Such a pointed map is a continuous map $X\to Y$, s.t. $x_0\mapsto y_0$.
Here are some important properties (just a few of them):


*

*If $x_0$, $x_1$ belong to the same path connected component, then $\pi_1(X,x_0)\cong \pi_1(X,x_1)$. (naturality)

*$\pi_1(id_{(X,x_0)})=id_{\pi_1(X,x_0)}$ and $\pi_1(f\circ g)=\pi_1(f)\circ \pi_1(g)$, which is just the functoriality and is easy to prove.

*We can define a category of pointed topological spaces and maps up to homotopy, in signs $\mbox{pTop/hom}$. In this context the following properties are given:


*

*If $f,g\colon (X,x_0)\to (Y,y_0)$ are homotopic, then $\pi_1(f)=\pi_1(g)\colon \pi_1(X,x_0)\to \pi_1(Y,y_0)$. The functor $\pi_1\colon \mbox{pTop}\to \mbox{Gps}$ factors through a functor $\bar \pi_1\colon \mbox{pTop/hom}\to \mbox{Gps}$. 

*If there are maps $f\colon (X,x_0)\to (Y,y_0)$ and $g\colon (Y,y_0)\to (X,x_0)$, s.t. $fg$ and $gf$ are homotopic to $id_X$ and $id_Y$ respectively, then $\pi_1(X,x_0)$ is iso to $\pi_1(Y,y_0)$. This is just a consequence of the previous bullet point.



I recommend you to read something about the universal property of the free product, the link between Galois theroy and fundamental groups and covering spaces in general. To learn more about the categorical language in the context of the fundamental group, please read May's "A Concise Course in Algebraic Topology", chapter 2 and tom Dieck's Algebebraic Topology, chapter 2.5. They both use more category theory.
