what is the direct limit of free group of rank $n$

Question

Is the direct limit of free group of rank $n$ the free group with countable generators? Does some one know a textbook which explain direct limit in detail and gives many examples? I only know its definition from wiki

Answer 1

Yes, it is if you have the right inclusions.

I'm assuming you're taking $F(\{x_1,...,x_n\}) \to F(\{x_1,...,x_{n+1}\}), x_i\mapsto x_i$ for instance.

Then if you have a reduced word $w$ on $\{x_n, n<\omega\}$ it's a finite word so it belongs to some $F(\{x_1,...,x_n\})$ and is reduced there too; so it belongs to the direct limit.

Now if it were equal to $1$ in the direct limit it would mean that there is some $m\geq n$ such that $w=1$ in $F(\{x_1,...,x_m\})$. But as the word is reduced this amounts to $w=\epsilon$.

Hence the direct limit is simply the set of reduced words on $\{x_n, n<\omega\}$, where none of them is $1$ except for the empty word; that is: the direct limit is the free group on countably many generators.

A more general way to see it is the following (for the readers with a bit of knowledge in category theory) : the direct limit is actually a colimit, and since the functor sending a set to the free group is a left adjoint, it commutes with the colimit. Hence $\varinjlim_n F(\{x_1,...x_n\}) = F(\varinjlim_n \{x_1,...,x_n\})$. But the inclusions have been weel chosen, so $\varinjlim_n \{x_1,...,x_n\} = \{x_n, n<\omega\}$ and this is exactly what we wanted.