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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.

Tensor_Product
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    It is very possible for non-homeomorphic spaces to have the same fundamental group. For instance, think of all the different spaces with trivial fundamental groups. For instance, $\Bbb R^n, [0,1]^n$ and for $n\geq 2, S^n$ spring to mind. Also, products are a category-teoretical thing, so simply saying that $\pi_1$ preserves or respects (finite) products is a good way of describing your property. – Arthur Sep 14 '17 at 06:49
  • @Arthur Why most of the books leave category theory ? What are advantages of studying some mathematical concept using category theory language. I think one prefers it because one can study things in most abstract way using category theory. – Tensor_Product Sep 14 '17 at 06:57
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    That is exactly why category theory is popular and considered powerful: Studying things via category theory is an abstraction. It helps in ignoring pesky details that only clutter proofs and it makes results transferrable between fields. However, some things you just can't define or prove using only category theory. Also, in the case of an introductory textbook in a given field, using time and energy on introducing category theory as well as the field itself might be seen as a waste of space, time and energy. – Arthur Sep 14 '17 at 07:02
  • @Arthur Sorry this is irrelevant ,Can you mention one or two good book about Category theory. One which you like the most. – Tensor_Product Sep 14 '17 at 07:05
  • I have never read a book about category theory. What I know, I've just picked up from mentions in passing in other books and reading on wikipedia. So I don't have a recommendation, sorry. – Arthur Sep 14 '17 at 07:07
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    @Tensor_Product Basic Category Theory by Leinster is good for basic category theory (it gets as far as showing you what adjunctions, representable functors, and limits are, and why they're the same thing). – Patrick Stevens Sep 14 '17 at 07:23
  • @PatrickStevens Thank you. – Tensor_Product Sep 14 '17 at 07:28
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    I will throw in, contrary to the remark of @Arthur, that the "abstraction" provided by category theory does nothing for me, personally. What I like about category theory is its practical applications, for example that it gives a systematic method for constructing isomorphism invariants: for every functor from one category C to another one D, all isomorphism invariants in D become isomorphism invariants in C. – Lee Mosher Sep 15 '17 at 12:56

1 Answers1

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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.

Daniel Bernoulli
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    Generalizing to the fundamental groupoid avoids the need for pointed topological spaces, and is arguably more natural. – Derek Elkins left SE Sep 14 '17 at 07:46
  • You are right. In the textbooks, I mentioned in my answer, the chapters deal with the fundamental groupoid. – Daniel Bernoulli Sep 14 '17 at 15:13
  • See https://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one for some advantages of the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, a concept not dealt with in the above mentioned books! – Ronnie Brown Sep 15 '17 at 17:41